So, who doesn't love getting confused by probability? I know I do. Let's just jump in right now without further ado.
A couple has two children. Assume that when a person gives birth, they have a 50% chance of having a boy and a 50% chance of having a girl. The couple tells you that at least one of their children is a boy. What is the probability that the other child is a boy?
This is a pretty simple puzzle in terms of calculations, but it causes a lot of confusion in a lot of people, so I thought it'd be fun to address. Since the chances of having a boy or a girl is 50/50, one would naively assume that the knowledge about the first child doesn't effect the probability of the other being a boy or girl. Thus, most people say the answer is 50/50.
But this is of course wrong. If the couple tells you that they have at least one boy, there is a 2/3 chance that the other child is a girl. Weird, right? Remember, this has nothing to do with correlations between children. We are assuming that all births are independent of each other. So, why is this so. The easiest way to figure it out is by examining the ways that a couple can have two children and finding their probabilities. They are as follows (B = Boy, G = Girl):
BB 25%
BG 25%
GB 25%
GG 25%
Each of these have an equal probability (50% * 50% = 25%). If the couple tells us that they have at least one boy, than all that they have done is eliminated the last way of having two kids; meaning that we ignore the GG combination.
Thus, the remaining combinations are:
BB
BG
GB
and they occur with equal probability. Of the remaining choices, two of them involve a boy and a girl, and the other involves two boys. Thus, it is twice as likely to have a boy and a girl than two boys. Thus, if a couple tells you that they have at least one boy, it means that 2/3's of the time, their other child is a girl. This is really a problem of semantics. Most of people's confusion comes from the idea of having "at least" one boy.
We get a different answer if we phrase the question in the following way: "A couple has two children. The youngest child is a boy. What is the probability of the sex of the other child?" Here, the answer is 50% boy and 50% girl. So, what's the difference? Again, it becomes clear if we list the possibilities:
BB
BG
GB
GG
If they tell us that the younger child is a boy, we are only left with:
BB
BG
and these have equal probability. Thus, the second child is a boy half the time and a girl the other half.
Interesting, no? This is a relatively simple problem. Maybe we'll get some harder ones in the future...
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Yes, it's an interesting problem, but I much prefer the 3-door game show puzzle, mostly because 99% of people (it's good to throw in an arbitrary and made up statistic in the middle of a probability problem, right?) get it wrong and don't understand why, even when you tell them the answer.
ReplyDeleteWhile I'm sure you know the problem and solution, I'll write the problem here for the sake of your many loyal readers.
So suppose you're on a game show, and you're presented with 3 doors to choose from, one of which has a new car behind it, the other 2 have goats (or whatever, the point is you don't want those doors). So you make your choice, but the host, instead of opening the door you chose, opens one of the other doors he knows the car is not behind. Then he present you with a choice, you can either keep your first choice or change your pick to the other unopened door. What should you do?
Most people come to this conclusion, at the opening of the door, the probability that the car is behind either remainding door is 50-50, so it really doesn't matter. And most will go one step further and say you should never second guess yourself so they'll stick with their original choice. Well those people are most likely (2/3 of the time) are going to driving a goat to work. The correct answer is that you should always change your pick. The odds of winning when you change your pick are 2 out of every 3 times.
Anyway, I'm bored at work and your problem reminded me of that problem and I thought it would make me sound smart. Enjoy.