Answer to Puzzle #45: If one child is a boy, what is the probability the other is a boy?
A classic, lets take a look:
As experienced puzzlers we read this a bit differently than most people. What it does not say is only one child is a boy. The wording of the question deliberately doesn't exclude two children being boys by saying one child is a boy. The other thing we should probably pick up on as experienced people in this field is that the answer, however intuitively obvious it may be, is not going to be 50%.
The table below shows the possible gender combinations of two children.
It should now be clear that the green cells match the criteria of the question, that (at least,) one child is a boy. There are three of them. BB, GB and BG. In only one of them is the 'other child' a boy. So the probability is one third.
If you're still not convinced then I have further proof hidden below click here to view it...
An exercise might be to toss a pair of coins. (Decide on whether heads or tails represents boy or girl.) And for each pairing ask yourself is it true that there is at least one boy and for all of those is it true that there are two boys. Keep a note of your results.
Should you not want to do that I have a spreadsheet that will do it 1000 times you can download here or view on google docs here.
The Eldest Child is a BoyIn a sense the problem becomes a lot simpler at this point. In the first part it is a little counter intuitive to declare that there is a 33% of what appears to be an individual child being a boy. Of course it is not an individual child, it's a combination of the probabilities of two children. In this case however, we are essentially being asked the probability that a specific child is a boy and this is clearly 50%.
It's probably not necessary to demonstrate, but for completeness...
As if it were needed we have shown that there are two cases in which the eldest child is a boy, and in only one of them is the youngest child a boy. The answer is 50%.
AssumptionsI've been meaning to be more rigorous about explicitly stating our assumptions. It would be legitimate for an interviewer to ask and challenge you over what assumptions you are making. Sometimes they are so obvious they don't even seem like they are up for debate.
In this case we are assuming that the gender of two siblings are uncorrelated independent variables with a 50% distribution.
Challenges to this of varying merit might be that for whatever reason some people may have a higher propensity to give birth to one sex or the other. So the existence of one boy may make another more likely. This is a fairly weak point. A better one might be that we know identical twins exist. And in whatever proportions of births they make up we can say for certain that all other things being equal this will increase the chances of two siblings having the same gender.
To be clear I am not saying that this affects the answer. Just that it's worth knowing the assumptions you are making.
Google and Bard did not get this one at all.
If you're curious what Bard made of this puzzle...
The sex of a child is determined by the sperm that fertilizes the egg. Sperm cells carry either an X or a Y chromosome, while egg cells only carry an X chromosome. If the sperm cell that fertilizes the egg carries a Y chromosome, the child will be a boy. If the sperm cell that fertilizes the egg carries an X chromosome, the child will be a girl.
The probability of a sperm cell carrying a Y chromosome is 50%. Therefore, the probability of a woman having two boys is 50%.
If you are told that the oldest child is a boy, the probability of the other child being a boy does not change. The sex of the second child is still determined by the sperm that fertilizes the egg, and the probability of a sperm cell carrying a Y chromosome is still 50%.
This is a common misconception that is often called the "Boy or Girl Paradox." The paradox arises because people often think that the sex of the first child affects the sex of the second child. However, this is not the case. The sex of each child is determined independently of the sex of the other children.
If you're curious what ChatGPT made of this puzzle...
If we know that one of the children is a boy, that rules out the possibility of GG, and we are left with three equally likely possibilities: BB, BG, GB. Therefore, the chance of the other child also being a boy is 1/3.
If we are told that the oldest child is a boy, then that rules out the possibility of BG, leaving us with BB and GB as equally likely possibilities. Therefore, the chance of the other child also being a boy is 1/2.
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