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# Some probability problems for you :)

Here are some problems that came up in a class. See if you can answer them. They're harder than they seem. [quote]Problem 1:
You meet a woman, ask how many children she has, and she replies "two." You ask if she has any boys, and she replies "yes." Now you know that at least one of her children is a boy. What are the odds that her other child is a boy?<hr></blockquote>

[quote]Problem 2:
Suppose youre on a game show, and youre given the choice of three doors: Behind one door is a prize; behind the others, nothing. You pick a door, say No. 1, and the host, who knows whats behind the doors, opens another door, say No. 3, which is empty. He then says to you, Ill let you switch to door No. 2 if you wish. Is it to your advantage to switch your choice?<hr></blockquote>
These are right out of Scientific American but they are well known besides that.

The second one is known as the Monty Hall problem because it's like the game show Lets Make a Deal. The issue at hand is that the conditional probability changes mid problem. You are given more information "half way" through.

<a href="http://www.io.com/~kmellis/monty.html" target="_blank">http://www.io.com/~kmellis/monty.html</a>
For the first one you have three possible states. BB, BG and GG. Prob of BB =.25, BG=0.5, GG=.25. So if you know that she has a boy then you have to renormalize the remaining states. BB=.333 and BG=.666 So I would guess there's a 2/3 chance that she has a girl? Right? That's off the top of my head.

[ 12-03-2001: Message edited by: Scott H. ]</p>
Here's another Monty Hall one for people who like Bayesian stats

<a href="http://astro.uchicago.edu/rranch/vkashyap/Misc/mh.html" target="_blank">http://astro.uchicago.edu/rranch/vkashyap/Misc/mh.html</a>

UofC people are rather good at this stuff.
Ah, I wasn't expecting some smarty-pants to come along and solve it before anyone gave the wrong answers.

In the Monty Hall problem, one of the things that confuses the issue is that when he actually did it on Let's Make a Deal, he would only offer the switch if they picked the right one to begin with. If they picked the wrong one, he would just open it up and say "sorry." If that's the case, it would be better to stay rather than switch.
The trick to the first one (as Scott H pointed out) is to see that there are not three possibilities (BB, BG, GG) but four possibilities (BB, BG, GB, GG).

The Monty Hall question, though, is a lot trickier than it first looks....
They're both pretty neat.Another well known one is the Birthday paradox.
[quote]Originally posted by sizzle chest:
<strong>
The Monty Hall question, though, is a lot trickier than it first looks....</strong><hr></blockquote>

I read about it in an old edition of MacFormat (some shareware program in about issue 32). I can't remember why, but they said there was a distinct advantage of changing your guess.

Amorya
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