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  • OT: Another probability question

    My nephew asked me an interesting probability question yesterday. It reminded me of this thread:

    http://bbs.homeshopmachinist.net/sho...ighlight=queen

    .. 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?
    I'm an abstract poet and I didn't even think I was.

  • #2
    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.

    Comment


    • #3
      Originally posted by Forestgnome
      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.
      "Patriotism is the last refuge of a scoundrel"

      Comment


      • #4
        "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

        Comment


        • #5
          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.

          Originally posted by jep24601
          Even with the probability given for how many lightening strikes per month, the probability for the remaining days does not change.
          Paul A.
          SE Texas

          Make it fit.
          You can't win and there is a penalty for trying!

          Comment


          • #6
            Originally posted by SmoggyTurnip
            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.

            Comment


            • #7
              Originally posted by rancherbill
              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"
              1601

              Keep eye on ball.
              Hashim Khan

              Comment


              • #8
                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?
                Last edited by ed_h; 06-08-2012, 12:33 AM.
                For just a little more, you can do it yourself!

                Comment


                • #9
                  Originally posted by ed_h

                  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.
                  Last edited by MichaelP; 06-08-2012, 02:01 AM.
                  Mike
                  WI/IL border, USA

                  Comment


                  • #10
                    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.
                    For just a little more, you can do it yourself!

                    Comment


                    • #11
                      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.
                      Last edited by MichaelP; 06-08-2012, 04:04 AM.
                      Mike
                      WI/IL border, USA

                      Comment


                      • #12
                        The solution

                        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.
                        I'm an abstract poet and I didn't even think I was.

                        Comment


                        • #13
                          Originally posted by J Tiers
                          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.
                          .

                          Comment


                          • #14
                            [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.

                            Comment


                            • #15
                              Originally posted by Mcgyver
                              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.
                              I'm an abstract poet and I didn't even think I was.

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