# A family has three children. What is the probability that at least one of them is a boy?

According to me there are $4$ possible outcomes:

$$GGG \ \ BBB \ \ BGG \ \ BBG$$

Out of these four outcomes, $3$ are favorable. So the probability should be $\frac{3}{4}$.

But should you take into account the order of their birth? Because in that case it would be $\frac{7}{8}$!

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BBG and BGG are each three times as likely as BBB or GGG. See this question and the comments: for probability, you need to also assign weights to the outcomes, not just count them. –  ShreevatsaR Mar 3 '14 at 9:14
For counting problems like this it is better to denote the three child as $A,B,C$ and counting each case of the form $A$ is male , $B$ is female, $C$ is female and so on… –  Riccardo Mar 3 '14 at 9:15
A mother who has given birth to 2 boys is statistically more likely to give birth to another boy, than to a girl. It seems the answers all seem to assume that P(B)=P(G)=½. This assumption is only approximately correct. –  gerrit Mar 3 '14 at 15:45
@Niharika: To understand why your logic is incorrect, imagine applying it to purchasing a lottery ticket: there are two outcomes and one is favorable, so your odds of winning would be 1/2. This is intuitively not true for the same reason your logic is not. –  Andrew Coonce Mar 3 '14 at 16:44
@gerrit Can you cite any research that supports your claim that a mother who has given birth to two boys is statistically more likely to give birth to another boy? –  David Richerby Mar 3 '14 at 20:41

The complement of at least one boy is all three girls

So, $P($ at least one boy$)=1-P(GGG)$

$=\displaystyle1-\left(\frac12\right)^3$

This is the de facto way of solving problems of Probability of at least one in case of Binomial Distribution like tossing a coin etc.

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It is usually easy to think in terms of the complement in these type of problems. Nice answer. –  Bennett Gardiner Mar 3 '14 at 10:05
So you are saying the order does matter right? –  gideon Mar 4 '14 at 8:16
@gideon, have you noticed anything like order in the answer? –  lab bhattacharjee Mar 4 '14 at 8:46
I'm asking as a newbie :) @labbhattacharjee your answer suggests P(GGG) is 1/8 (which it is if there are possibilities, which will have to include order like DonAntonios answer) –  gideon Mar 4 '14 at 8:49
@gideon, no, order is immaterial here. I don't think DonAntonio has meant any order, he has listed the combinations –  lab bhattacharjee Mar 4 '14 at 8:51

There are in fact eight possible outcomes:

$$GGG\,,\,GGB\,,\,GBG\,,\,BGG\,,\,BBB\,,\,BBG\,,\,BGB\,,\,GBB$$

Of these, only one does not include a boy (B) in the event, and thus the probability of all girls is $\;\dfrac18\;$ .

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This assumes, of course, that the probability of male and female births is the same. The 2012 (most recent available) data for England and Wales published by the UK's Office for National Statistics shows a bias towards male births (51.3% of live births). I would expect the statistics for other regions to differ slightly from 50:50. –  DMM Mar 3 '14 at 11:49
@DMM Yes, of course. It could be that upon checking one single casino and a very fair, well-balanced die, the probability to get some number (say, 3) is slightly higher than $\;1/6\;$ (because of the hand throwing the dice, the table, the material the die is made of, etc.), but we usually assume some events to have "the usual, standard" probability to happen. –  DonAntonio Mar 3 '14 at 12:11
@DMM you could just as easily say that the probability is $50\%$ and it there was an experimental error of $2.6\%$ ;) –  Sabyasachi Mar 3 '14 at 18:47
@DonAntonio Except that every measurement everywhere shows that more males are born than females. Yes, you can arbitrarily declare some ratio the "standard" probability, but why? –  Kundor Mar 3 '14 at 20:25
@Sabyasachi reminds me of my favourite joke from The Big Bang Theory. "(story about asking a physicist for help with his chickens that won't lay eggs) Okay, I have a solution, but it'll only work for spherical chickens in a vacuum!" –  Cruncher Mar 4 '14 at 21:32

Another way to look at this is to draw this out

Here I follow the stereotypical association of gender and colors: the blue boxes represent boys and the pink boxes represent girls. Each time you have a boy or a girl, in the next generation you can have a boy or a girl also, so the number of possibilities is doubled each generation.

In terms of your problem, when you have a boy, that represents a checkmark against "at least one of them is a boy", so I've crossed the box concerned. However all the subsequent generations after this boy are also families in which there is at least one boy, so I've crossed those out too. You can see that the chance of having at least one boy is $1/2$ in the first generation, $3/4$ in the second, and $7/8$ in the third. This generalizes to $(2^n-1)/2^n$ in the nth generation.

Conversely the chance of having no boys is $1/2$ in the first generation, $1/4$ in the second, and $1/8$ in the third. This generalizes to $1/2^n$ in the nth generation.

(Essentially I've drawn a probability tree diagram here, which generalizes to much more complicated problems).

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Should your second-last sentence be "the chance of not having at least one boy"? –  Andrew Leach Mar 4 '14 at 18:37
@AndrewLeach thankyou, brain dzzzzt –  TooTone Mar 4 '14 at 18:39

I’m adding an answer since I’m not reputable enough to add a comment to “lab bhattacharjee”’s answer. There are some assumptions to consider here:

Assuming that the family is generated the old fashioned way then “lab bhattacharjee”’s answer is mostly correct with only a few more explicit assumptions, as follows.

1. Assuming that the slight natural skew towards human male progeny is ignored
2. Assuming adoption is not under consideration
3. Assuming modern medical reproductive techniques are not being used, including...
4. Assuming no selective abortion, then only...
5. Assuming we are counting the children and not the parents. Because given that the father is male, that makes the “real” answer to this question, as given by “Niharika” and edited by “Ric Ped”, out to be 100% and no statistics are needed
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The possibilities are

ggg ggb gbg bgg gbb bgb bbg bbb

at least one boy... 7/8

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this is a duplicate of an existing answer math.stackexchange.com/a/697515/77151. Did you miss it? –  TooTone Mar 4 '14 at 18:42

This is a combination question, not permutation. The possible combinations are BBB, BBG, BGG, GGG. The formula is (n+r-1)!/r!(n-1)!, where n = the number of possible types to choose from (boy or girl) = 2 and r = the number of elements = 3. Probability of this result is 3 out of four or 75%.

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I think you misunderstood the question. –  Cameron Williams Mar 3 '14 at 17:41
Thanks Cameron. I understood the question but forgot to express the answer properly. –  user132797 Mar 3 '14 at 19:29
@user132797 In the sense that you forgot to express the answer as $\tfrac78$ instead of $\tfrac34$? I think you misunderstood the question. –  David Richerby Mar 3 '14 at 19:42
It doesn't matter whether order is brought up or not; you have non uniformly removed possible outcomes from the search space. In short, your answer presumes each of those outcomes is equally likely when they are in fact not. –  UpAndAdam Mar 3 '14 at 22:14
Yes! Your possible combinations would be HHH, HHT, HTH, HTT, THH, THT, TTH,TTT. While some of these have the same number of heads, it does not follow that they are the same combination. –  Amy Mar 3 '14 at 22:57

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