There is a family that has 3 kids. One of them is a boy. What is the probability that the youngest is a boy?

Answer 1

ambiguous. did you mean
a) exactly one is a boy
b) at least one is a boy
c) a specific child John is a boy.

a) 1/3
b) of the eight possible combinations of boys and girls, one (ggg) is eliminated.
So each of the others has a 1/7 chance of happening.
these consist of 1 with 3 boys, 3 with 2, and 3 with one.
So the chances of the youngest being male are
1/7(1 +3(2/3) +3(1/3)) = 4/7

c) if John is the youngest 1. If not the information is useless and the probability is 1/2. 1/3 chance that John is youngest.
1/3(1) +2/3(1/2) = 2/3

Answer 2

if the family has 3 kids and only one is a boy, the probability that the youngest is a boy is 1/3.

if the family has 3 kids and we know one is a boy, but we don't know the gender of the other two, then it becomes complicated.

I'll use B for the one kid we know and put them in birth order
Bgg
Bbg
Bgb
Bbb
bBg
bBb
gBg
gBb
bbB
bgB
ggB
gbB

Here you see that by knowing the gender of one child, the odds become 2/3 that the youngest is a child. The odds are 1/2 for the 2/3 of the time that the one you know is not the youngest and 100% for the 1/3 of the time when it is. That works out to 2/3 overall.

Answer 3

Consider 8 families, each with 3 kids. The genders are uniformly distributed, so we have one family with boy-boy-boy, one with boy-boy-girl, ... one with girl-girl-girl. That covers all the possibilities.

Only one of these families has has all girls, that leaves 7 with "one is a boy". Of these 7 families, 4 have "youngest is a boy".

The probability is 4/7.

Answer 4

1 out of 2.

These are independent events. The outcome of the first two births does not affect the outcome of the third birth.

Answer 5

Depends on whether the other two are girls or boys.

Answer 6

one in two