What is the probability of getting at least 2 boys out of 3 children?

Answer 1

As asked - this question has no calculable answer as it depends on the genetics of the parents but ...

If we assume, to make the problem a little easier, that the odds of a boy are equal to the odds of a girl, then p(boy)=p(girl)=.5 Now, at least two boys means odd of 2 or 3 boys out of three children.

The formula for finding k successes, (let's call a boy a success ONLY because that is what we are looking for) out of n trials (births) is P(n,k)=n!/(n-k)!k! (p)^k(q)^n-k ( note that the first part of this formula is 3Cn where n is 2 or 3 in this case.

So P(2,3)=3x2x1/2x1(.5)^2(.5) =3(.25)(.5)=.375 and P(3,3)=3x2x1/3x2x1(.5)^3(.5)^0 =.125 Now add these two probabilities .375+.125=.5

You could also just do a tree, with just the three children:

B.G

/ \

B.G B.G

/ . | . | . \

B.G B.G B.G B.G

The first row is the first child, then if it's a boy, look to the left half. If the first one is a girl, look to the right half. So for 3 boys, follow the very left-hand path (B-B-B).

There are 4 possibilities that yield 2 or more boys (out of the 8 total possibilities). These are {BBB, BBg, BgB, gBB}. I bolded the bottom-row ones that are the path to 2 or more boys. Probability = 4/8 = 0.5, or 50%. Doing this validates that the first method is correct (the graphical method becomes unwieldy when more events are added).

One important rule to use in any problem that asks at least is the complement rule.

The idea is simple. P(No boys )+ P( 1 boy) + P(2 boys) + P(3 boys)=1 since those are all the possible outcomes. We subtract P( No boys) and P(1 boy ) from both sides and we have

P(at least 2)=1-P(none)-P(1). In this case no boys means GGG and the chance of that is

(1/2)^3 or 1/8. Also one Girl can be GBB, BGB or BBG and each has a 1/8 chance so the three together have a 3/8 chance. This means P( as least two boys)= 1-1/8-3/8 or 1-4/8=4/8 or 50%

The "at least method is very useful when looking at large numbers of outcomes. For example, chance of at least 1 boy in 100 children. The tree would be hard to make and the binomial formula would take a long time to use. However 1-P(none) would give you the answer in a minute. It is 1-(1/2)^100.