From families with four children, a child is selected at random and found to be

Question

From families with four children, a child is selected at random and found to be a girl. What is the probability that she has an older brother? Assume that in a four-child family all sex distributions are equally probable.

Hint: Let G be the event that the randomly selected child is a girl, A be the event that she has an older brother, and O, U, L, Y be the events that she is the oldest, upper middle, lower middle, and the youngest child of the family, respectively. For any subset B of the sample space let Q(B) = P (B|G); then apply Theorem 3.3(Law of Total Probability) to Q.

Explanation / Answer

Sol:

Here there are 4 positions to fill, that is, there are 4 children, and two possible choices, boy and girl.

So total possible combinations are: 2^4 = 16

A child is selected at random and found to be a girl.

Let A denote the event that this randomly selected girl child has an elder brother.

P(she has an older brother) = 1 - P(she does not have an older brother)

Now,

P(she does not have an older brother) is calculated by considering the possible cases.

Case 1: She is the youngest child.

In this case, there are 3 higher level positions to fill. But there is only one case in which event A cannot occur,

which is, when we have all elder three children as girls.

So total possible combinations for this case = 1

Case 2: She is the second youngest child.

In this case, there are 2 higher level positions to fill and 1 lower level position. The possibility of event A not

occuring can occur in following scenarios:

So total possible combinations for this case = 1

Case 3: She is the second oldest child.

In this case, there are 1 higher level position to fill and 2 lower level positions. The possibility of event A not  

occuring can occur in following scenarios:

So, total possible combinations for this case = 1+1 = 2

Case 4: She is the oldest.

In this case, there are 3 lower level positions. The possibility of event A not  

occuring can occur in following distinct scenarios:

So, total possible combinations for this case = 1+1+1 = 3

So total ways in which event can not occur = 1+1+2+3 = 7

So,

Probability that event A does not occur = 7/16

So,

Probability that event A occurs = 1 - 7/16 = 9/16 = 0.5625

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