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SOOPOO
SOOPOO
Joined: Aug 8, 2010
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October 19th, 2013 at 2:48:09 AM permalink
On August 8, 2010, I proposed essentially this very question, which is my favorite 'simple problem'. I titled the thread, SIMPLE PROBLEM. It goes something like this.... Assumption... boys and girls are born with equal frequncies.

Question.... A woman has two children, (not identical twins). One of them is a boy, what are the odds that the other one is a boy? Of course the answer is 1/3, but you can imagine that virtually everyone you ask will blurt out 1/2.
Dalex64
Dalex64
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October 19th, 2013 at 9:26:30 AM permalink

I finally figured it out in a way that I understand, to get the answer of 2 in 3.
You pull a coin with the white side facing up out of the bag.
you are either looking at:

the only white side of the white/black coin. the bottom is black.
one white side of the white/white coin. the bottom is white.
the other white side of the white/white coin. the bottom is white.
ThatDonGuy
ThatDonGuy
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October 19th, 2013 at 2:18:41 PM permalink
Quote: SOOPOO

On August 8, 2010, I proposed essentially this very question, which is my favorite 'simple problem'. I titled the thread, SIMPLE PROBLEM. It goes something like this.... Assumption... boys and girls are born with equal frequncies.

Question.... A woman has two children, (not identical twins). One of them is a boy, what are the odds that the other one is a boy? Of course the answer is 1/3, but you can imagine that virtually everyone you ask will blurt out 1/2.


That's because the question can be interpreted in different ways. If, by "one of them is a boy," you are referring to a specific child (e.g. the older of the two), then the answer is 1/2. This is what causes the Monty Hall Problem to be so confusing - the intended answer assumes that, when you say, "Monty opens a door and shows (something besides the grand prize)," his choice of door was arbitrary (to be sure it would not be the prize) and not random.

Also, when you say, "One of them is a boy," it can be inferred that "only one of them is a boy," in which case the answer is zero.
dwheatley
dwheatley
Joined: Nov 16, 2009
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October 19th, 2013 at 2:38:19 PM permalink
I was asked this question during an interview at Goldman Sachs.
Wisdom is the quality that keeps you out of situations where you would otherwise need it
pacomartin
pacomartin
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October 20th, 2013 at 6:52:21 AM permalink
The question "What is the probability that you have chosen a two color coin or a single color coin?" is compelling, but irrelevant. But it is not the question.
SOOPOO
SOOPOO
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October 20th, 2013 at 7:37:06 AM permalink
Quote: ThatDonGuy

That's because the question can be interpreted in different ways. If, by "one of them is a boy," you are referring to a specific child (e.g. the older of the two), then the answer is 1/2. This is what causes the Monty Hall Problem to be so confusing - the intended answer assumes that, when you say, "Monty opens a door and shows (something besides the grand prize)," his choice of door was arbitrary (to be sure it would not be the prize) and not random.

Also, when you say, "One of them is a boy," it can be inferred that "only one of them is a boy," in which case the answer is zero.



I don't know why you would think when I say 'one of them is a boy' that would in any way infer I was talking about a specific child. Quite the opposite, that is why I specifically used the words I did. And if I meant the facts to be that "only' one of them was a boy, I would have used the word 'only'.
I stand by the question.... and the answer....
Wizard
Administrator
Wizard
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October 20th, 2013 at 8:16:18 AM permalink
Quote: MrCasinoGames

I think To make the puzzle more fun and more difficult, the question may be put down as:
There are 3 coins in a bag. One is white on one side and black on the other. The other 2 is white on both sides.



That is the way many people tell it. However, it is easy to eliminate the all-black coin as a possibility. Adding it seems to be unnecessarily adding useless information.

Quote: SOOPOO

Question.... A woman has two children, (not identical twins). One of them is a boy, what are the odds that the other one is a boy? Of course the answer is 1/3, but you can imagine that virtually everyone you ask will blurt out 1/2.



It is asked that way a lot too. Actually, it is a slightly different problem. Both can still be solved simply using Bayes.

If you go this way I think you should make it more clear it is a random woman drawn out of the population of all women with two children, except those with two girls.
It's not whether you win or lose; it's whether or not you had a good bet.
pacomartin
pacomartin
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October 20th, 2013 at 9:01:08 PM permalink
Quote: Wizard

I think you should make it more clear it is a random woman drawn out of the population of all women with two children, except those with two girls.



I would say that the answer is 1/2 if you don't specify that statement.
MathExtremist
MathExtremist
Joined: Aug 31, 2010
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October 20th, 2013 at 9:27:44 PM permalink
Quote: pacomartin

I would say that the answer is 1/2 if you don't specify that statement.


Isn't it technically neither 1/2 nor 1/3 but based on the actual gender split at the time the question is being asked? It's a qualitatively different question than the coin question. In the coin question, you know the whole population. But in the child question, you're assuming that the probabilities for a woman having two children (in order) are p(boy,girl) = p(girl,boy) = p(girl,girl), which is almost certainly not true.
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
kubikulann
kubikulann
Joined: Jun 28, 2011
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October 21st, 2013 at 6:50:07 AM permalink
Quote: Wizard

You randomly draw one coin, and observe one side only, which is white.

To play the lawyer: it does not say how you choose the side you observe. Everybody assumed it was at random. But it could be a "Monty Hall"-like problem, where the operator deliberately chooses the white side to show to you, in the event you drew the bicolor coin.

Then the answer is not the same.
Reperiet qui quaesiverit

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