PDA

View Full Version : OT: Another probability question

SmoggyTurnip
06-07-2012, 03:49 PM
My nephew asked me an interesting probability question yesterday. It reminded me of this thread:

.. so I thought some of might find it interesting.

Suppose that in a given month there is an equal likelyhood of a lightning strike on any given day. In other words if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p.

Now suppose that lightning DID strike on the first day of the month.

The question:
Which of the remaining days of the month has the greatest probability of being the next day that lightning strikes?

Forestgnome
06-07-2012, 04:36 PM
Given that there was no probability given for how many lightening strikes per month, the fact that you were struck on one day does not change the probability for the remaining days.

jep24601
06-07-2012, 04:59 PM
Given that there was no probability given for how many lightening strikes per month, the fact that you were struck on one day does not change the probability for the remaining days.

Even with the probability given for how many lightening strikes per month, the probability for the remaining days does not change.

cameron
06-07-2012, 05:19 PM
"Suppose that in a given month there is an equal likelyhood of a lightning strike on any given day."

There is a serious ambiguity in this statement. Suppose the liability (or probability) of a strike on any given day is P. Then, in a 30 day month, the probable number of strikes is 30xP. Which would put us back into a problem somewhat like the first red queen problem. Which, I suspect, is not what you intended by this statement. Or, at least, not what many would interpret the meaning to be.

Dave Cameron

Paul Alciatore
06-07-2012, 05:30 PM
As stated, the original problem was, "Suppose that in a given month there is an equal likelyhood of a lightning strike on any given day." There is nothing in the original problem that says that the probability will change based on the actual events in previous days. So, as stated below and by others, the probability for the successive days does not change even if lightning has struck on one, two, or all of the previous days. This is basic to the way in which the problem is stated.

This is an often misunderstood idea in probability theory.

Now, if you were dealing from a deck of cards, then the probability of getting an ace would decrease when it is known that one or more aces have already been dealt. This is because there are fewer aces left in the deck.

Even with the probability given for how many lightening strikes per month, the probability for the remaining days does not change.

RancherBill
06-07-2012, 09:46 PM
Suppose that in a given month there is an equal likelyhood of a lightning strike on any given day. In other words if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p.

Now suppose that lightning DID strike on the first day of the month.

The question:
Which of the remaining days of the month has the greatest probability of being the next day that lightning strikes?

This is too vague. I was watching the 'Weather Office' and there were 2000+ strikes per hour.

J Tiers
06-07-2012, 10:24 PM
This is too vague. I was watching the 'Weather Office' and there were 2000+ strikes per hour.

You are being very literal...... I read the problem as "Equal.....strike in a particular place on any given day"

ed_h
06-08-2012, 12:30 AM
I believe we can do this without getting too deep in the statistics.

If the probability that there is no strike in a particular day is P(0), then the chance of at least one strike is 1-P(0). These numbers are the same for every day.

The chance of a strike on day 2 is 1-P(0).

The chance that the next strike is on day 3 is the probability of no strike on day 2 times the probability of at least one strike on day 3: P(0)*(1-P(0))

For day 4 it is: P(0)*P(0)*(1-P(0)).

Since P(0) is less than 1, it appears that the successive probabilities will get smaller and smaller. This implies that the highest probability for the next strike is on day 2.

Is there an error here?

MichaelP
06-08-2012, 01:46 AM
These numbers are the same for every day.
++++++++++++++++++
The chance that the next strike is on day 3 is the probability of no strike on day 2 times the probability of at least one strike on day 3: P(0)*(1-P(0))
+++++++++++
it appears that the successive probabilities will get smaller and smaller. This implies that the highest probability for the next strike is on day 2.

Is there an error here?

I don't follow your logic. And you contradict yourself. Or maybe it's too late, and I cannot think straight... However, there is an equal chance that I'm simply dumb. :)

The probability of lightning is equal any day of the month. This is given. Build you logic and math on this fact.

ed_h
06-08-2012, 02:40 AM
But the question isn't

"What is the probability that lightning will strike on day n?",

it is

"which day has the highest probability of having the first strike (after the initial one)?"

Having the first strike on day 5 requires that days 2 through 4 have no strike, and the probability of that is less than 1.

MichaelP
06-08-2012, 03:59 AM
I see what your drift is. In order to be the first day with a strike, all previous days should go without any strike. Since one day without a strike is more probable than two or more consecutive days without strikes, the more days there are between the strikes, the less the probability is.

It makes sense. But only as long as it doesn't contradict the given conditions. And the given conditions are that the likelyhood of strikes is exactly the same every day. Who knows why. Maybe the clouds become denser and more energized every single day and that compensates for the expected decrease of probability. Whatever it is, the result is the equal probability every single day. And that's not only the conditions, but also the answer to the OP question.

IMHO, of course.

SmoggyTurnip
06-08-2012, 08:54 AM
ed_h nailed it.

I will just restate what he said here using the symbols that were given in the problem. And for those that have little or no experience with probability theory just remember that probabilities are expressed as a number between 0 and 1. So an event with a probability of 1 is an event that is certain to happen and an event with a probability of 0 is an impossible event. And an event with a probability of .5 will happen half the time (ie. Getting heads in a coin toss).

In our question the probability lightning strikes on the 2nd day is p, the same as any other day. So the probability or the 2nd day being the next day that a strike occurs is p.

In order for the 3rd day to be the NEXT day that a strike occurs lightning must NOT strike on the 2nd day. To calculate the probability of 2 events happening in sequence we multiply their probabilities together. And the probability of no strike on day 2 is (1-p). so we have

The probability of the next strike occurring on the 3rd day is (1-p)*p
The probability of the next strike occurring on the 4th day is (1-p)(1-p)*p
...
The probability of the next strike occurring on the Nth day is p*(1-p)^N

So you can see that the probability of any day being the next day that a strike occurs is (1-p) times the probability that the day before it was the next day that a strike occurs. Since (1-p) is less than 1 the probabilities for each day being the next day of a strike continue to get smaller and approach 0.

This may makes perfect sense if you think about it in the following way:

Suppose we replace the idea of a lightning strike with flipping heads in a coin toss, and you flip a coin once every day from now until eternity passes. Luckily you flip heads on the first day. Well it is clear that you have the exact same likely hood of getting heads on any given day (.5) but the probability that the NEXT day that you get heads is 30 days from now is pretty much 0 as you would have to flip tails every day for 29 days – that aint gonna happen.

It appears to me that this problem is pretty close to the same problem as the 1st red queen problem.

Mcgyver
06-08-2012, 09:23 AM
You are being very literal...... I read the problem as "Equal.....strike in a particular place on any given day"

would it matter? whether the boundary is the surface of the earth or sq cm there is a probabilty between 0 & 1 of it hitting there... but it will not be 0 or 1 so you still have a probability to fuel the question.

Forestgnome
06-08-2012, 10:13 AM
[QUOTE=SmoggyTurnip]The question:Which of the remaining days of the month has the greatest probability of being the next day that lightning strikes?[QUOTE]

You can read this to mean "On day two, which of the remaining days of the month has the greatest probability of being the next day that lightning strikes?" It's a question of the probability on one specific day.

SmoggyTurnip
06-08-2012, 10:40 AM
would it matter? whether the boundary is the surface of the earth or sq cm there is a probabilty between 0 & 1 of it hitting there... but it will not be 0 or 1 so you still have a probability to fuel the question.

Actually it could be considered 1. For example if you select the whole earth the probability of a lightning strike some where on the earth is 1 because lightning strikes somewhere on earth thousands of times a day every day. In this case it will strike on the first day of the month, and the probability of the second day of that month being the next day that lightning stikes is also 1. Which agrees with our calculations because the probability of the 3rd day being the next day of a strike is 0 same for the 4th 5th etc. The only possible Next strike day is the second day.

If you select 1 square centermeter of the earth the probability of a lightning strike there is very close to 0 but the reasoning will still hold if there is strike on that spot on the first day of the month. If you do the math for any probability greater than zero you will always find that the second day of the month is the most likely day to have the NEXT strike.

MichaelP
06-08-2012, 07:51 PM
...the probabilities for each day being the next day of a strike continue to get smaller and approach 0.

And how does it work with the main condition you stated: "...if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p" ? :)

Maybe your question should've been composed differently?

ed_h
06-09-2012, 12:30 AM
The question

"What is the probability of a strike on day n?"

and the question

"What is the probability that day n has the next strike?"

are different questions, and have different answers.

MichaelP
06-09-2012, 12:43 AM
True. :):)

RancherBill
06-09-2012, 01:33 AM
You are being very literal...... I read the problem as "Equal.....strike in a particular place on any given day"

You have fallen for the vague troll's question, as have I.

The question is vague!!!!!

Suppose, we are talking about a mountain top. It has two trees, one is hit and is reduced to ashes. The probability that location (tree) being hit again is 0.000000. The other tree will be hit.

The other posters can do all their math, but it aint gonna happen - the second tree will be hit. Thus it will not happen in the same location.

+ or - Zero
06-09-2012, 01:50 AM
My nephew asked me an interesting probability question yesterday. It reminded me of this thread:

.. so I thought some of might find it interesting.

Suppose that in a given month there is an equal likelyhood of a lightning strike on any given day. In other words if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p.

Now suppose that lightning DID strike on the first day of the month.

The question:
Which of the remaining days of the month has the greatest probability of being the next day that lightning strikes?

If the probability was equal for each day and a strike did occur on day one then a strike will occur on every day.

When it struck on day one it proved the probability of a strike was 100% on day one. As all days have the same probability, so each day must follow the next at the now established 100% probability of a strike.

Else the probability for each day would have to be different for each day.

Q.E.D.

ed_h
06-09-2012, 02:53 AM
"...a strike will occur on every day."

Wow. That's some logic.

+ or - Zero
06-09-2012, 03:28 AM
"...a strike will occur on every day."

Wow. That's some logic.

If an equality is stated in any logic question then the statement becomes provable (or disprovable) within the stated equal terms.

This question is nothing more then "If A=B and B=C, then C=A" stated in an intentionally confusing way.

So if p=p for each day, and p=100% on one day, then each day has a p of 100%. Else the statement of equality was not true.

As we are required to use the stated equality as true, the only condition under which it can be true, as stated, is if each day has a lightning strike.

MichaelP
06-09-2012, 04:34 AM
When it struck on day one it proved the probability of a strike was 100% on day one. If an event happened, it doesn't mean its probability was 100%. It only means it was above zero.

P.S. Your screen name matches the topic perfectly. :)

+ or - Zero
06-09-2012, 04:52 AM
If an event happened, it doesn't mean its probability was 100%. It only means it was above zero.

P.S. Your screen name matches the topic perfectly. :)

Then what it means is that every day has exactly the same "above zero probability" that = a lightning strike.

If p=p for each day then any event happening on any day must happen on all days or the days are not of equal probability for the event.

Go study a bit of logic.

+ or - Zero
06-09-2012, 05:52 AM
Where the confusion is happening is in the notion of what probability means and how that relates to an equality.

Lets look at it as coin tosses:

Everyone is most likely familiar with the statement that the heads/tails probability of any given coin toss is 50% for either heads or tails coming up, and that no mater how many heads or tails have come up before it does not effect the next toss still being a 50/50 probability.

But for any given toss, once the toss is made the probability is not 50/50 for that toss --it's been decided and is 100% either heads or tails.

So the little word problem about lightning strikes relies on the reader thinking that each day has some unknown probability for a lightning strike. Something like the coin toss, i.e., each new day is a new event not constrained by any past event.

But they slide in a statement of equality for the probability of the event happening on each day --something the exact opposite of the coin toss, i.e., each new day is no longer a new event with an unknown probability for a given event to happen (lightning strike), it is an event that you have been slyly told has a certain and definite probability, not an unknown at all, for when they state a strike did occur on day one the probability for day one was 100% for a lightning strike. Just like the 100% probability that a coin toss has *after* it's been made.

Because they say each day has an equal probability of a strike, and you have been told (if surreptitiously) that day one has already had a strike (making that days probability of a strike 100% --it happened, you can't have less then a 100% probability for an event happening when the event has already happened) every day that follows must then have an exactly equal probability of a strike as the first day (100%, equal to the first day) --that is what equal means.

Adding that each day has an equal probability to an event happening, that has already happened makes the probability of each consecutive event have an equal probability of happening as the one that has already happened --is the exact opposite from the coin toss probability where each event has a probability unrelated to any other coin toss event.

Read the above slowly if you don't grasp it right away. Don't feel bad if it's difficult to see the misdirection in the puzzle --that's why it's a puzzle, it is supposed to misdirect your thinking.

Mcgyver
06-09-2012, 09:30 AM
Actually it could be considered 1. For example if you select the whole earth the probability of a lightning strike some where on the earth is 1 because lightning strikes somewhere on earth thousands of times a day every day.

It's not 1, its .9999 with a lot nines afterwards. There is the possibility, however remote, that no lightening will strike the planet tomorrow, anywhere.

My intent wasn't to push it to the ridiculous, only to point out that the question implies consistency in the area within which lightening must strike and that's all that is required because any size area will have a probability greater than zero and less than 1.

ed_h answered it in the 8th post, the next day has the highest probability.

Is there an error here?

I dont think so....day 2 is P, day 3 is P x probability it wouldn't strike on day 2 etc etc

Mcgyver
06-09-2012, 09:55 AM
You have fallen for the vague troll's question, as have I.

The question is vague!!!!!

Suppose, we are talking about a mountain top. It has two trees, one is hit and is reduced to ashes. The probability that location (tree) being hit again is 0.000000. The other tree will be hit.

The other posters can do all their math, but it aint gonna happen - the second tree will be hit. Thus it will not happen in the same location.

So just take the question to mean it strikes anywhere on the mountain, or the county. it does not matter so long as you are considering the same area for each day. Ignore any physical alternations the lightening made that might change probabilities, its noise - this is a math question not a lightening prediction tool.

ed_h
06-09-2012, 11:14 AM
Read the above slowly if you don't grasp it right away.

Its a pretty straightforward statistics problem.

When a ststistical event of this type happens to occur, it doesn't change the probability if its future occurance.

If you are really claiming otherwise, here is where we ask you to provide a credible source that supports your opinion.

ed_h
06-09-2012, 11:25 AM
Suppose, we are talking about a mountain top. It has two trees, one is hit and is reduced to ashes. The probability that location (tree) being hit again is 0.000000. The other tree will be hit.

Folks, I believe this was intended to be a question about statistics, not a question about lightning.

jep24601
06-09-2012, 12:32 PM
Kind of reminds me when I was watching the news one evening and the weatherman came on and said " there's a 60% chance of rain and - oh! it's happening right now"

+ or - Zero
06-09-2012, 01:22 PM
Its a pretty straightforward statistics problem.

When a ststistical event of this type happens to occur, it doesn't change the probability if its future occurance.

If you are really claiming otherwise, here is where we ask you to provide a credible source that supports your opinion.

OK, I'll do that. This sort of problem has been being hashed out since around 1927 when Werner Heisenberg stated his Uncertainty principle http://en.wikipedia.org/wiki/Heisenberg_uncertainty_principle

In 1935 Austrian physicist Erwin Schrödinger devised a thought experiment, sometimes described as a paradox http://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat

Here is the actual Schrödinger equation, from around 1926: http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

Now you may start your studies with those links, and/or you may consider this:

All groups of statistical events describe a probability curve of some shape --you are possibly aware of the bell curve that students are sometimes graded against each other with.

However a probability curve can take any shape that the probability of any given event, or group of events, happening may have, it is not restricted to a bell shape.

The shape of the probability curve for an event that has already happened (been observed, i.e, the wave has collapsed) is a straight line, no curve at all.

The paradox presented in the example we are discussing comes from the statement that all probabilities are equal within the framework stated (30 days), and that one of those events has already happened.

So when you plot the statistical probability curve you are constrained to plotting each consecutive day such that it's probability is equal to the day before, because that first day has a known probability, which was 1 (the event happened --no curve on day one, just a straight line, or more precisely a single point plotted on a line that must be straight if extended, as it is the only sample at that point).

So day two comes along and you want to see where the probability of the event happening may be plotted on your probability curve... but wait, in our little paradox it must be equal to every other day in our set of days. And as we have already plotted one of those days our new point on our curve must be equal to the prior day... lather, rinse repeat.

Our probability curve, for the constrained set of daily events we must plot, turns out to be a straight line, as all 30 point we plot must be equal on our probability curve plot to the first point we plotted.

It would also help you to study a bit of Set theory http://en.wikipedia.org/wiki/Set_theory which dates back to 1874 in a paper by Georg Cantor.

What we have in our little 'set' of events is a cardinality of 1 for the first event --the single event on day one. The a statement that all days in a greater set (30 days) must have the same cardinality (be equal to the cardinality of the event on day one of the greater set).

This causes the probability of the event happening on every day of the greater set to also have a cardinality of 1, as they all must be equal.

Here's what you need to study about the cardinality of sets (and yes a set may consist of one thing, a number or an event, or many such things): http://en.wikipedia.org/wiki/Cardinal_number

And so, as I said to start with, lightning strikes each day for the 30 days.

Q.E.D.

I have tried to keep this explanation as simple as possible for you, but you may want to consider that you are in fairly deep waters here, it takes considerable study to really get into this stuff. But it is worth it, if you have the time and inclination --there are wonderful things to learn out there, if one will but look for them.

ed_h
06-09-2012, 03:49 PM
...you may want to consider that you are in fairly deep waters here, it takes considerable study to really get into this stuff.

Your condescending tone is misplaced here. I'm not sure how you think Heisenberg or Shrodinger are remotely applicable. I suspect clumsy name dropping.

Most of the rest of your post seems to be empty googlectual bluster and red herrings.

A citation to support your claim would normally include a text and section or page number. A credible web site with a reference to a specific passage would be even better, so we could all check it and maybe learn somethng new.

Mcgyver
06-09-2012, 04:02 PM
I'm left wondering how anybody but the first guy ever won a lottery :D

+ or - Zero
06-09-2012, 04:52 PM
Sorry if that's the way you take it, I never claimed to have mastered the art of telling someone they are wrong in a way that they liked.

I guess I left out the link that might have helped you most:

Because that is the crux of the problem as stated --it is an apparent paradox because it requires you to believe that two (or more --30 in this case) apparently independent events are equal to each other, which would mean they are in fact related events not independent from each other.

It is an apparent paradox because there is one condition where both statements can be true --if, and only if, lightning strikes happen on each day.

You have fallen into the trap set by the apparent paradox in assuming only that "When a ststistical event of this type happens to occur, it doesn't change the probability if its future occurance." (spelling errors yours) is relevant to an equation containing a statement of equality.

This problem is just a form of the well known birthday problem, with the twist that every birthday has to be the same, i.e., equal.

The result is, of course, that everyone must have the same birthday to make the problem as stated a true statement.

And here's a link (one of many) that discuss that form of probability, among others.
http://130.37.52.27/tijms/sample.pdf

In the shortest way I can say this is;

The problem was :

'Suppose that in a given month there is an equal likelyhood of a lightning strike on any given day. In other words if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p.

Now suppose that lightning DID strike on the first day of the month.

The question:
Which of the remaining days of the month has the greatest probability of being the next day that lightning strikes?"

The give away is here:
"In other words if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p"

The probability on day one was clearly 100% as a strike did happen --might or might not have happened if less the 100%, but it did happen so it was a 100% probability that it would happen.

Before an event happens it's p is whatever it may be, but once you define an event as having happened it becomes a certainty that it did happen, that becomes a p of 100% for that event.

Therefore the p of the strike on the Nth day is also the same. Exactly as stated in the problem.

I really am sorry that my style of attempting to show a point is offensive, it really is not meant that way. For me one should have faith in his convictions and show that strongly in his actions and words.

Now you may think what ever you wish from here on. I have conducted class all I want to --you are now wasting my time.

Good luck in your endeavors, and maybe we'll agree on something else some other time.

+ or - Zero
06-09-2012, 05:51 PM
I'm left wondering how anybody but the first guy ever won a lottery :D

Relax, someone wins the lottery everyday, it's a 100% probability that someone will win another lottery, even after the first person or persons won the last lottery. Each lottery a separate and unique event, you know.

Unfortunately the probability that it will be you or I that wins is considerably lower... ;)

ed_h
06-09-2012, 06:42 PM
The probability on day one was clearly 100% as a strike did happen...

A single credible citation to support this claim. Just one.

You can skip the convoluted gibberish.

A googled link to an entire site or page on statistics isn't support.

Help us learn.

MichaelP
06-09-2012, 09:18 PM
Apparently, if we follow the logic of the fine gentleman, probability of any event can be either 100% or zero: the event either happens or doesn't happen.

OK. I'm out of here (went "to learn a bit of logic").

+ or - Zero
06-09-2012, 09:52 PM
A single credible citation to support this claim. Just one.

You can skip the convoluted gibberish.

A googled link to an entire site or page on statistics isn't support.

Help us learn.

OK, one last try. Please read the entire post including the citations. And at the end I have provided a more pop version of how past and future can interact in the way I'm am trying to show --and by someone much better at it then I am.

http://mathworld.wolfram.com/Event.html
An event is a certain subset of a probability space. Events are therefore collections of outcomes on which probabilities have been assigned.

http://mathworld.wolfram.com/Outcome.html (http://mathworld.wolfram.com/Outcome.html)
An outcome is a subset of a probability space. Experimental outcomes are not uniquely determined from the description of an experiment, and must be agreed upon to avoid ambiguity (Papoulis 1984, pp. 24-25).

http://mathworld.wolfram.com/ProbabilitySpace.html
A triple (S,S,P) on the domain S, where (S,S) is a measurable space, S are the measurable subsets of S, and P is a measure on S with P(S)=1.

http://mathworld.wolfram.com/ProbabilityMeasure.html
Probability Measure
Consider a probability space specified by the triple (S,S,P), where (S,S) is a measurable space, with S the domain and S is its measurable subsets, and P is a measure on S with P(S)=1. Then the measure P is said to be a probability measure. Equivalently, P is said to be normalized.

http://mathworld.wolfram.com/Probability.html (http://mathworld.wolfram.com/Probability.html)
Probability
Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%. The analysis of events governed by probability is called statistics.

There are several competing interpretations of the actual "meaning" of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation), while Bayesians treat probability more subjectively as a statistical procedure that endeavors to estimate parameters of an underlying distribution based on the observed distribution.

A properly normalized function that assigns a probability "density" to each possible outcome within some interval is called a probability density function (or probability distribution function), and its cumulative value (integral for a continuous distribution or sum for a discrete distribution) is called a distribution function (or cumulative distribution function).

A variate is defined as the set of all random variables that obey a given probabilistic law. It is common practice to denote a variate with a capital letter (most commonly X). The set of all values that X can take is then called the range, denoted R_X (Evans et al. 2000, p. 5). Specific elements in the range of X are called quantiles and denoted x, and the probability that a variate X assumes the element x is denoted P(X=x).
....
And more you can go read it if you want.

The part we care about is that there are no variables in our little problem.

It is just "there are 30 p 's, and all p 's = p and, event=strike, and event happened in in p=1". No variable is stated at all, so all probabilities are equal to the first p wherein an event happened. So nothing can vary from p=1 --all p 's being equal they all must contain an event.

Further consider that (as from Wolfram above) " In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%."

We have been told (in the problem at hand) that the probability of a event in p=1 is 100% because we are told it did happen, it is a certainty and equal to 100%.

If Wolfram is not a citation to your liking then I suggest Brian Greene's book "The Fabric of the Cosmos":
http://www.pbs.org/wgbh/nova/physics/fabric-of-cosmos.html

Or watch the full four hour video version on the web here:
http://richarddawkins.net/videos/645228-the-fabric-of-the-cosmos-with-brian-greene-watch-the-complete-nova-series-online

Go for the second hour:
http://www.pbs.org/wgbh/nova/physics/fabric-of-cosmos.html#fabric-time
Fabric: The Illusion of Time, to get right into the pop version of the hard science I've been talking about.

Brian Greene is much better at explaining stuff then I am, but he is also much more general and avoids any actual hard stuff like I have tried to point you at.

Enjoy the show.

ed_h
06-10-2012, 12:34 AM
Yes, I'm pretty familiar with the Wolfram site, including some of those pages.

However, just posting links to statistics or pop science sites online does not lend support to your case, and none of your more focused links seemed to address this issue.

You've made a pretty specific claim about about the way probabilities work. This would require a pretty specific citation to support it, like to a specific paragraph in a credible source. This should be easy, right?

Just the citation, so we can read it. The long winded posts don't help us (or you).

+ or - Zero
06-10-2012, 12:42 AM
Yes, I'm pretty familiar with the Wolfram site, including some of those pages.

However, just posting links to statistics or pop science sites online does not lend support to your case.

You've made a pretty specific claim about about the way probabilities work. This would require a pretty specific citation to support it, like to a specific paragraph in a credible source. This should be easy, right?

Just the citation, so we can read it. The long winded posts don't help us (or you).

Well you don't have to read them, and I will not bother anymore. To me it seems a case of "There are none so blind as those that will not see." You have my sympathy.

As I said once, good luck in your future endeavors, and possibly we will find something to agree on one day. But it obviously won't be this day.

ed_h
06-10-2012, 12:58 AM
...won't be this day.

Probability of THAT is 100%

Jaakko Fagerlund
06-10-2012, 04:42 AM
Suppose that in a given month there is an equal likelyhood of a lightning strike on any given day. In other words if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p.

Now suppose that lightning DID strike on the first day of the month.

The question:
Which of the remaining days of the month has the greatest probability of being the next day that lightning strikes?
The 2nd day.

The probability of a strike is above 0 because of the fact that lightining DID strike on the first day, so it is above zero and equal or less than 1.

"+ or - Zero" gets it true in an event that the probability is 1, but doesn't take in to account that with a probability between 0 and 1 there can be lightning.

If a day has 50 % chance of lightning, and lightning happens, the probability doesn't change, it is still 50 %.

-----

So, if strike occurs on the first day, the second day has the highest probability of striking the next time. If the next lightning happens on a later day, it means that the previous days (days 2...n-1) have had no lightning and this would mean a lower probability.

So, the answer is 2nd day.

+ or - Zero
06-10-2012, 06:23 AM
The 2nd day.

The probability of a strike is above 0 because of the fact that lightining DID strike on the first day, so it is above zero and equal or less than 1.

"+ or - Zero" gets it true in an event that the probability is 1, but doesn't take in to account that with a probability between 0 and 1 there can be lightning.

If a day has 50 % chance of lightning, and lightning happens, the probability doesn't change, it is still 50 %.

-----

So, if strike occurs on the first day, the second day has the highest probability of striking the next time. If the next lightning happens on a later day, it means that the previous days (days 2...n-1) have had no lightning and this would mean a lower probability.

So, the answer is 2nd day.

Very close, the probability is the same on every day, so says the problem... "if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p."

So if P-1 is a strike event, then the probability on day one is 100%, as you have stated.

But that means that because the statement is that all days have the same probability (any Nth day has the same probability as any other Nth day), and the first day was 100%, then in order for them all to have an equal probability, they all must have the same 100% probability as on P-1, else the statement that "if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p." could not be a true statement.

The required conditions inherent in the "if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p." must be met or it isn't a true statement. The only way to satisfy the stated condition is if all days have the same probability as the first day --100%.

We have, after all, been told that all Nth days have the same probability, and p-1 is an Nth day, like any other Nth day in the set.

This is not really about probability as such, it's about "Is there a condition where all the requirements of the initial statement can be true?" It's a logical true or false question. The probability issue as stated in the problem amounts to a straw man and that is where much confusion comes from.

In order for it to really be an issue of probabilities it would require some variable to have some probability of happening or not happening. As the event did happen on day one, and each day must be of equal likelihood to the first day there is nothing left open to be a true variable.

I know that seems counter intuitive, but that's the nature of statements contrived in such a manner as to mislead. That's why they are brain teasers.

yul m6
06-10-2012, 10:04 AM
+ or minus
"if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p."
Let's do it in terms of a coin toss: "If the probability of heads on the first toss is 50% then the probability of heads on the Nth toss is also 50%"
Do you agree??
Now we will put your statement in terms of a coin toss: "If the probability was equal for each day and a strike did occur on day one then a strike will occur on every day", so it becomes:
"If the probability was equal for each toss and heads did occur on the first toss then heads will occur on every toss".

Now I realize the meaning of convoluted logic.

Mcgyver
06-10-2012, 10:20 AM
He's assumed that the probability changes after the event, that if you get a TAILS, the probability of tails in that instance is no longer 50/50 but one. This seems an erroneous assumption.....but I think he's reading it like a riddles vs probability question. ie since i'm telling you lightening strike that day it must be 100%. I read the question as here's the scenario, then as it turns out lightening did strike the first day etc

I've read and listened to lectures by those physicists mentioned, they or +- offer no indication as to how quantum mechanics enters into this or why in a macro event has its probabilities change. If the chance of lightening striking the first day was 1/1,000,000, whether it strikes or not does not change the probability of lightening striking that first, it is still 1/1,000,000.

imo where he fell of the rails was assuming P for D1 must equal 1 because the event happened.

As to all the condescending language, instead, perhaps +- could offer his credentials in the area so might have a better sense as to whether this is the oddest presentation to ever grace these pages or in fact he's several orbits above as and we're just not getting it :D

+ or - Zero
06-10-2012, 10:27 AM
+ or minus
"if the probability of a strike on the first day is p then the probability of a strike on the Nth day is also p."
Let's do it in terms of a coin toss: "If the probability of heads on the first toss is 50% then the probability of heads on the Nth toss is also 50%"
Do you agree??
Now we will put your statement in terms of a coin toss: "If the probability was equal for each day and a strike did occur on day one then a strike will occur on every day", so it becomes:
"If the probability was equal for each toss and heads did occur on the first toss then heads will occur on every toss".

Now I realize the meaning of convoluted logic.

The heads or tails part creates an actual variable that has some degree (in the case of a or b variables with equal probability, 50%), but the lightning strike is a single event --the event happens or not, but if not then the statement (p=p) is not a true statement.

The event does not and can not have two possible states when it happens, as a coin toss does, it happens or it doesn't but the possibility of it not happening was removed in the initial statement wherein it did happen.

The coin toss analogy predicts the result of an event or a series of events where each event may have a different outcome, it does not have any given probability of the event it's self (the toss) happening or not.

Now if you said that for 30 tosses in a row the toss would happen then you would have a rough analogy of what is really going on in the problem as stated.

+ or - Zero
06-10-2012, 10:34 AM
As to all the condescending language, instead, perhaps +- could offer his credentials in the area so might have a better sense as to whether this is the oddest presentation to ever grace these pages or in fact he's several orbits above as and we're just not getting it :D

Wasted effort, as everyone knows (or should know) "On the Internet, nobody knows you're a dog" --Peter Steiner published by The New Yorker on July 5, 1993

I'd post the cartoon here but it may be copyrighted. But it's an easy find, and funny.

+ or - Zero
06-10-2012, 03:55 PM
The 2nd day.

The probability of a strike is above 0 because of the fact that lightining DID strike on the first day, so it is above zero and equal or less than 1.

"+ or - Zero" gets it true in an event that the probability is 1, but doesn't take in to account that with a probability between 0 and 1 there can be lightning.

If a day has 50 % chance of lightning, and lightning happens, the probability doesn't change, it is still 50 %.

-----

So, if strike occurs on the first day, the second day has the highest probability of striking the next time. If the next lightning happens on a later day, it means that the previous days (days 2...n-1) have had no lightning and this would mean a lower probability.

So, the answer is 2nd day.

You know Jaakko, I actually owe you an apology.

Because I've been trying to show why a strike occurs on every day, I slid over this part of the question "greatest probability of being the next day that lightning strikes?"

Clearly if a strike occurs every day, day two is the "next day" that lightning strikes. With 100% certainty in so far as my position about strikes is concerned.

So you actually have the correct answer as to which day is going to be the next day with a strike.

But it is correct because all days have strikes, and that was what I was focused on.

SmoggyTurnip
06-11-2012, 09:26 AM
If the probability was equal for each day and a strike did occur on day one then a strike will occur on every day.

When it struck on day one it proved the probability of a strike was 100% on day one. As all days have the same probability, so each day must follow the next at the now established 100% probability of a strike.

Else the probability for each day would have to be different for each day.

Q.E.D.

If you take this position then there are only 2 possible values for p. 0 and 1. Which would pretty much makes the whle field of practical probability useless. p in these types of problems in a measure of our ability to predict weather an event will occur or not. It has nothing to do with with weather the event will occur or not. When a coin is tossed in the air the outcome is already determined by the laws of physics and it is not a random event but we say it has a probability of .5 because we can't predict the outcome one way or another. Someone else who took some fancy measurements and could do some quick calcs could come up with a different number such as .8. Both would be correct as they are reporting the measure or their aboilty to predict the outcome.

The problem as stated only implies that our ability to predict a strike on each day is the same (p). The fact that that there was a strike on day 1 does not change our ability to predict the outcome on day 2 or any day for that matter. (In the problem as given).

If you want to say that p=1 on day one since there was a strike then we are not measuring the same thing and thus you do not understand the question.

+ or - Zero
06-11-2012, 01:31 PM
If you take this position then there are only 2 possible values for p. 0 and 1. Which would pretty much makes the whle field of practical probability useless. p in these types of problems in a measure of our ability to predict weather an event will occur or not. It has nothing to do with with weather the event will occur or not. When a coin is tossed in the air the outcome is already determined by the laws of physics and it is not a random event but we say it has a probability of .5 because we can't predict the outcome one way or another. Someone else who took some fancy measurements and could do some quick calcs could come up with a different number such as .8. Both would be correct as they are reporting the measure or their aboilty to predict the outcome.

The problem as stated only implies that our ability to predict a strike on each day is the same (p). The fact that that there was a strike on day 1 does not change our ability to predict the outcome on day 2 or any day for that matter. (In the problem as given).

If you want to say that p=1 on day one since there was a strike then we are not measuring the same thing and thus you do not understand the question.

And you are taking it as the event equals the outcome. They are two different things. Please see the Wolfram definitions I provided.

When you take a set of events that have no outcome (they simply happen or not) and then say the likelihood of the event happening is equal for 30 days, whatever event happened (or did not happen) on any day has to be the same for all days.

This is because the events are not probabilistic within the framework provided in the question, they are deterministic --they are determined by the question it's self (there shall be 30 of them, all equal).

What happens here is that in the coin toss which is probabilistic because each event (the toss) has a possibly different outcome (heads/tails), then the series of tosses which are usually normalized to 'as many as to show heads/tails have a probability of .5 for each event', but may (and often are) stated as a limited number of tosses and then compared to a yet larger number of tosses in order to show that just becase in a set of say 10 events even though the outcome may be 8/10 in favor of heads, in a larger set of events, the probability will always drift towards 5/10 heads (or tails). But in all cases the number of events is not what determines the outcomes (heads/tails), because the event and it's outcome from one toss to the next is independent.

Now I know you knew that, but what I also see is that you are not considering that the question as stated is about how many events happen, not about any outcome from those events --and the event either happens or not, just as the coin toss it's self --the toss, either happens or not. All the question really does is tell you that an event will happen 30 times and the likelihood of the event happening is equal on all days.

It therefore follows that what ever event happened on any day, all days must also have that event happen, else they would not be equal for the occurrence of that event.

Events by them selves either happen or not, it is the outcome of an event that has the probability factor, not the event.

You can indeed devise questions about the probability of events happening, but this question is not one of them. It is internally deterministic as to the events happening when it states that all events (not outcomes, for there is no outcome from the event(s) in the question) are the same, i.e. equal. It is actually just a statement about how many events will happen, and it tells you the particular event it is referring to (strikes).

As a statement it can be true or false, for it to be a true statement there must be a strike each day.

edited for typo.

ed_h
06-11-2012, 02:51 PM
This thread still alive? I thought it petered out a couple of days ago.

In the original question, let's remove the part that says that lightning struck on the first day.

It's the morning of day 1, and lightning may or may not strike today. If it does strike, does that mean it must strike every day this month?

If yes, then I'm afraid I can't help any more.

If no, then let's move to the morning of day 2. The situation is now identical to the original problem.

So is it yes or no?

+ or - Zero
06-11-2012, 03:38 PM
This thread still alive? I thought it petered out a couple of days ago.

In the original question, let's remove the part that says that lightning struck on the first day.

It's the morning of day 1, and lightning may or may not strike today. If it does strike, does that mean it must strike every day this month?

If yes, then I'm afraid I can't help any more.

If no, then let's move to the morning of day 2. The situation is now identical to the original problem.

So is it yes or no?

Actually when you remove the part about the strike happening on the first day, then indeed it does become a question of probability about strikes on any given day, and as the probabilities are stated as being equal for each day, then each day has a .5 probability of a strike.

And the part of the question that asks which day has the highest probability of a strike becomes meaningless.

That is because you change the question from being deterministic as to all events can only be equal if a strike happens each day (the event has occurred in one day, and the be equal must then occur in each day), to making each day a separate event in and of it's self --there is no condition stated as to the outcome of the question over the 30 days in which isolated events may happen or not. Each day may then have it's own outcome.

In other words you change the question into a valid question about probability, you make the event 'happen/not happen' become the outcome of the question. The outcome(s) of events can be expressed as having probabilities.

I really hope you might consider this issue again, perhaps I am 'zero-ing' in on a way to express the problem in a way easier to understand. At least I hope so.

Edit: Actually I should have said that each day has some equal probability of a strike --the fact that a strike may occur on one of the days or none of the days is not considered in the revised question, so all days could have strikes, or no days, or something between those constraints of nature, So it is not really .5 it's just an unknown but equal probability for any day. I tend to suspect that if they are equal probabilities then .5 is correct for the given set, but have not given that issue much thought. The question about the day that has the highest probability of a strike remains meaningless either way in the revised question, as they are all stated as equal, whatever the actual probability is.

ed_h
06-11-2012, 03:46 PM
"It's the morning of day 1, and lightning may or may not strike today. If it does strike, does that mean it must strike every day this month?"

So is the answer to the question yes or no?

+ or - Zero
06-11-2012, 04:12 PM
"It's the morning of day 1, and lightning may or may not strike today. If it does strike, does that mean it must strike every day this month?"

So is the answer to the question yes or no?

I'm not going to respond to you if you keep trolling. That's what brought on the talking down to you to start with. Which I am sorry about, I should handle trolls better.

ed_h
06-11-2012, 04:55 PM
Not sure how you can consider this a troll.

According to your view of how probabilities work, is the answer to the question yes or no?

To wit:

It's the morning of day 1, and lightning may or may not strike today. If it does strike, does that mean it must strike every day this month?

+ or - Zero
06-11-2012, 05:07 PM
Not sure how you can consider this a troll.

According to your view of how probabilities work, is the answer to the question yes or no?

To wit:

It's the morning of day 1, and lightning may or may not strike today. If it does strike, does that mean it must strike every day this month?