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 you?re on a game show, and you?re 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 what?s behind the doors, opens another door, say No. 3, which is empty. He then says to you, ?I?ll let you switch to door No. 2 if you wish.? Is it to your advantage to switch your choice?<hr></blockquote>
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 you?re on a game show, and you?re 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 what?s behind the doors, opens another door, say No. 3, which is empty. He then says to you, ?I?ll let you switch to door No. 2 if you wish.? Is it to your advantage to switch your choice?<hr></blockquote>
Comments
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>
[ 12-03-2001: Message edited by: Scott H. ]</p>
<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.
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 Monty Hall question, though, is a lot trickier than it first looks....
<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