SOLUTION: Please help on this probability question. I got 1/16 as an answer by multiplying (1/2)*(1/2)*(1/2)*(1/2) = 1/16 I am not sure if I am right. A couple decides to have 4 children.

Algebra ->  Probability-and-statistics -> SOLUTION: Please help on this probability question. I got 1/16 as an answer by multiplying (1/2)*(1/2)*(1/2)*(1/2) = 1/16 I am not sure if I am right. A couple decides to have 4 children.       Log On


   



Question 1047218: Please help on this probability question. I got 1/16 as an answer by multiplying (1/2)*(1/2)*(1/2)*(1/2) = 1/16 I am not sure if I am right.
A couple decides to have 4 children. If they succeed in having4 children and each child is equally likely to be a boy or a girl, what is the probability that they will have exactly 2 girls and 2 boys?

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2F2%29%2A%281%2F2%29%2A%281%2F2%29%2A%281%2F2%29+=+1%2F16
is the probability that they will have
girl, girl, girl, girl, in that order.
It is also the probability of
boy, boy, boy, boy, in that order,
and it is the probability of any sequence of 4 children,
such as
boy, boy, girl, girl, or
girl, girl, boy, boy, or
boy, girl, boy, girl, or
girl, boy, girl, boy, or
boy, girl, girl, boy, or
girl, boy, boy,girl.
So out of the 16 sequences possible
there are 6 sequences that would give them exactly 2 girls and 2 boys.
The probability of that is 6%2F16=highlight%283%2F8=0.375=%2237.5%25%22%29 .
When you are faced with problems of that sort,
think of %28boy%2Bgirl%29%5E4 or %28a%2Bb%29%5E4 .
In the painful %28a%2Bb%29%5E4=%28a%2Bb%29%2A%28a%2Bb%29%2A%28a%2Bb%29%2A%28a%2Bb%29 multiplication,
you would end with 2%2A2%2A2%2A2=2%5E4=16 products,
including red%284%29 a%5E3b product terms, green%286%29 a%5E2b%5E2 , and blue%284%29 ab%5E3 ,
but then you "collect like terms" to get
.
The term green%286%29a%5E2b%5E2 tells you that there were green%286%29
outcomes where 2 a's and 2 b's happened, in various orders.
The probability of that happening is green%286%29%2F16 .