# A woman has 2 children. If 1 is a boy, what are the odds of her next child being a girl?

I know it sounds simple, but the odds of her having Two boys in her lifetime are 1/2*1/2=1/4. Therefore, if the first time, she has a boy, the odds of her having another one are only 1/4. So aren't the odds of her having a girl the second time 3/4?

Relevance

Before having any children, or even getting pregnant, we assume that no information is known, (no babies of definite gender are yet specified), so we base our estimates pf probable events on a tendency to match statistical averages or probabilities based on X and Y chromosomes, but there is no guarantee that actual events will correspond.

Consider the comparable situation of tossing coins. If heads and tails are regarded as equally likely, then after two tosses the results could be HH, HT, TH or TT so the odds of having two heads seems to be 1/4. But the coins do not have to obey the probabilities strictly. It often occurs with a short number of tests that you get a run of three tails for example, or any other unbalanced ratio of heads to tails. This is because those probabilities calculated only begin to apply for a large number, (such as thousands of tests). Then the ratio heads/tails usually tends to get nearer to 1/1.

By the way, having had a run of 7 tails, it is not more likely that you will get an 8th tail. Neither is it true to think that you have “used up” all your tails chances and that the next time you are more likely to get a head. The probability is 50:50 in the long run, which means that in as much as we can say anything at all, heads and tails are equally likely at the start of each toss. (This assumes the coin never lands on its edge of course).

Returning to the maternity ward, you will know that biological science can nowdays influence how many and what gender of child are to be born, but I will assume that in your calculations you wish to ignore such possibilities.

Before getting pregnant we are in a land of almost no information and the best we can say, (with the reservations expressed above), is that the odds of her having 2 boys in her lifetime, (assuming she chooses to have children in the first place and then does not change her mind about having a second after finding out about how often thev first one cries), are calculated on the sparse basis 1/2*1/2 = 1/4 as you said.

Following your discussion, suppose the first child is born and it is a boy. We now have new and definite information that trumps our original rough guess. The possibility that the first child will be a girl is now to be calculated as zero, so the original basis of this calculation is now defunct. If we ignore interference by biological science, the odds of her next child being a girl should be about 1 in 2.

Humans are more complicated than coins. Consider a culture where having a female child was considered undesirable. Of course we would not approve of that, But if the mother’s incorrect calculations gave her to think that the odds of her having a boy the second time were 3/4, it could be that when the father heard this he would more readily agree to their having a second child; and so the miscalculated probability could be said to have a sort of feedback effect, making a second birth and any chance of boy or girl more likely.

Regards – Ian H

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• You ask for odds, but in your details, you only talk about probabilities. Odds and probability are not the same thing.

The probability that her next child is a girl is 1/2. The odds are 1:1 that the next will be a girl. The number of childern she currently has and their gender has nothing to do with the gender of her next child.

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• Anonymous
5 years ago

if woman has one boy and is expecting another then probability to be a girl is 1/2 if didnt consider the first birth or 1/4 if we consider it • Problem Solved
Lv 5
5 years agoReport

This is exactly what I thought

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• Still one half as it starts all over again.

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