OT: probability calculations

[MattiJ]

I have 3 cups upside down and one of them has prize inside.
You choose one of the cups and I lift one of the cups you didn't choose and reveal what is inside.

Cup was empty, now there is two cups left. What is the probability that the cup you selected is the prized cup?
Should you change your selection at this point and what is the probability if you change your selection at this point?

This was related to coronavirus testing but we can skip that part..

[mklotz]

Monty Hall problem...

https://en.wikipedia.org/wiki/Monty_Hall_problem

[MattiJ]

Spoiler :P

[ed_h]

Yes, it's the well known and much discussed Monty Hall problem. The answer is non-intuitive to most.

Ed

[QSIMDO]

Aw, c'mon, I'm still struggling over Schrodinger's Cat.

[Mcgyver]

Now if you had three cats in three boxes and you opened one and the cat was ____________ .......

[dian]

such stuff is not the main problem of statistics. the main problem of statistics is that they are statistically insignificant in almost all cases. (since you mention corona testing.)

[lynnl]

The Monty Hall problem starts off as a probability question, and then the solution offered is mainly a psychology one.
A classic case of obfuscation!

[ed_h]

The Monty Hall problem starts off as a probability question, and then the solution offered is mainly a psychology one.
A classic case of obfuscation!

No, not at all. The solution is objective, provable, and testable.

Ed

[Paul Alciatore]

Not if the host's choices are either truly random or if they follow a fixed pattern.

I suspect that those choices are not even the actual host's choice. Having been behind the scenes in many TV productions, I would suspect that he is told which door to open by someone behind the scenes. And that choice can be made in any of several ways. This could also be conveyed to him, on the fly, in a number of ways. For instance, in any TV production there is always a person who is called a Floor Director. This is a person who is on the studio floor, somewhere between the cameras and who is connected to the Director in the control room with a headset. He/she literally conveys the actual show director's directions to the host and others in the program, usually with hand signals that they watch for. That Floor Director could simply hold up one, two, or three fingers to signal which door to open first. If that is the case, then the show's host really does not need to even know which door is the winning one. That would prevent the host from giving any unintentional signals to the contestant. And that is how I would set it up.

The Monty Hall problem starts off as a probability question, and then the solution offered is mainly a psychology one.
A classic case of obfuscation!

[ed_h]

The relationship between the statistics problem and the TV show is pretty incidental. Someone applied the name of the show to the problem because it is a real world example, but the problem would still exist even if the show never had.

Ed

[Erich]

The problem boils down to one of two choices.

First, Pick a door any door. There is a wonderful prize behind one door and a goat behind the other two.
Now Here is you second choice. Would you trade your door for BOTH of the other doors?

Hmmm Do I want one chance to win or two chances to win? I know, give me the second choice. I want two chances to win. The fact that one of the doors not chosen at first is shown to have a goat is irrelevant. You knew that going in. Because there are either two goats in them or one goat.

[J Tiers]

I expect this is leading up to the problem of antibody testing. Much is made of antibiody testing, and "coronavirus passports" that show the person is (maybe) now "immune" so they are safe and supposedly are also not a danger to others and can move about freely (of course we do not know that is possible, but never mind that).

Suppose you have a test that is 95% accurate, that is it ALWAYS identifies actual antibodies, but it gives a false positive 5% of the time for people who actually have NO antibodies. So only 5% of the tests on unexposed persons will suggest that they WERE exposed.

That seems pretty good, pretty good accuracy, and it should be good to go, right?

Now 1000 people take the test, and suppose 3% of them have been exposed. So you have 30 people who show up with antibodies, and 970 people who do not have antibodies.

So what do you get as a result?

You get 78 people identified as being "safe", 48 of whom are totally unprotected but would be allowed to go around freely, catching the virus and spreading it. You actually have more false positives than true ones.

So the test that "looks pretty good", is in fact pretty nearly useless.

Even if 10% had been exposed, still almost 1/3 of the "OK" people would actually be totally unprotected.

This would be different if a LOT of people were exposed, then the test would be reasonably workable.

[elf]

The problem boils down to one of two choices.

First, Pick a door any door. There is a wonderful prize behind one door and a goat behind the other two.
Now Here is you second choice. Would you trade your door for BOTH of the other doors?

Hmmm Do I want one chance to win or two chances to win? I know, give me the second choice. I want two chances to win. The fact that one of the doors not chosen at first is shown to have a goat is irrelevant. You knew that going in. Because there are either two goats in them or one goat.

What if you really wanted a goat because you were tired of mowing your lawn?

[dian]

I expect this is leading up to the problem of antibody testing. Much is made of antibiody testing, and "coronavirus passports" that show the person is (maybe) now "immune" so they are safe and supposedly are also not a danger to others and can move about freely (of course we do not know that is possible, but never mind that).

Suppose you have a test that is 95% accurate, that is it ALWAYS identifies actual antibodies, but it gives a false positive 5% of the time for people who actually have NO antibodies. So only 5% of the tests on unexposed persons will suggest that they WERE exposed.

That seems pretty good, pretty good accuracy, and it should be good to go, right?

Now 1000 people take the test, and suppose 3% of them have been exposed. So you have 30 people who show up with antibodies, and 970 people who do not have antibodies.

So what do you get as a result?

You get 78 people identified as being "safe", 48 of whom are totally unprotected but would be allowed to go around freely, catching the virus and spreading it. You actually have more false positives than true ones.

So the test that "looks pretty good", is in fact pretty nearly useless.

Even if 10% had been exposed, still almost 1/3 of the "OK" people would actually be totally unprotected.

This would be different if a LOT of people were exposed, then the test would be reasonably workable.

thank you for explaining what i said in post #7.