What is the probability that she has 3 boys?

Question

A woman has three children and there is an equal probabilty of each being male/female, i.e. P(male)=P(female)=.50.

The woman tells you that she has a least one boy. What is the probability that she has 3 boys?

Answer 1

The answer is 1/7.

Use Bayes Theorem, which states:

P(A|B) = P(B|A) P(A) / P(B)

in this case,

P (3 boys | at least 1 boy)
= P(at least 1 boy | 3 boys) P(3 boys) / P(at least 1 boy)
= (1) * (1/8) / (7/8)
= 1/7

Answer 2

Ricky D and n0body give correct answers. The rest are giving you wrong answers. But consider the following:

This problem is a classic, because it is an example of how the answer depends on how one interprets the woman's statement (that she has at least one boy).

The simplest way to view it is that there are 8 possible gender patterns for 3 children, which are as follows (B = boy; G = girl):

BBB
BBG
BGB
GBB
BGG
GBG
GGB
GGG

Of these 8 possibilities (which we can think of as 8 different families), any of the first 7 mothers could have made the statement that she had at least one son. Out of those 7, one has 3 boys, so the probability of 3 boys would be 1/7.

But here's another interpretation:
Out of these 8 mothers, you chose one at random. If the chosen mother has only boys, she says, "I have at least one son." If she has only girls, she says, "I have at least one daughter." and if she has at least one of each, she flips a coin to decide whether to say she has a son or that she has a daughter.

In this example, there are 6 mothers who have at least "one of each." And on average, 3 of them would (after the coin flip) say, "I have at least one son." Now there are only 4 mothers making that statement, and 1 of those 4 has 3 sons, so now the probability is 1/4.

And you can probably come up with other scenarios for what lies behind the mother's statement, leading to other probabilities.

If you have no other information, try 1/7. But the problem is ambiguous. And it's a well-known type of ambiguous probability problem.

Answer 3

What is the probability that she has 3 boys?

Answer 4

Each of these eight possibilities (for first, second, and third child) is equally possible

1. BBB
2. BBG
3. BGB
4. BGG
5. GBB
6. GBG
7. GGB
8. GGG

You know that it cannot be No.8. Hence there are just seven possibilities left, and the only one that has 3 boys is 1. Hence the probability is 1/7.

maybe!

Answer 5

All are wrong, except Ricky D (did not see his answer when I started writing).
"At least one boy" doesn't mean he is first. There are more possibilities, but you don't have to enumerate them all.
It is easier to look at it this way. There are total of 8 combinations (2^3).
Knowing that there is at least one boy invalidates 1 - three girls. Out of the remaining 7 we need one. Thus the probability is 1/7.

Answer 6

Since she has at least one boy the following are the only possibilities: bbb, bbg, bgg. So the answer would be 1/3.

Answer 7

I had implantation bleeding for one day?

Answer 8

Why isn't the queens husband a king?

Answer 9

GIVEN that she KNOWS 1 is boy then the answer is
(1)(1/2)(1/2) = 1/4

and NOT (1/2)(1/2)(1/2) = 1/8

Answer 10

possibilities

BGG
BBB
BGB
BBG

probability: 25%