Question

A family has four children whose gender is unknown. If we randomly select one child from this family and find out that she is a girl, what is the probability that she is the only girl in this family

Answer 1

The answer = 4/16

Explanation:

The probability of having a boy is 1/2 and the probability of having a girl is 1/2

Probability that they will be the same gender (all males) =1/2 x 1/2 x 1/2 x 1/2 = 1/16

Probability that they will be the same gender (all females) =1/2 x 1/2 x 1/2 x 1/2 = 1/16

The probability of not having a girl = 1/16

The probability of having at least a girl = 1 - (The probability of not having a girl)

=1- (1/16)

=15/16

Method 1:

Using tree diagram for better explanation:

{g,b,b,b}

{g,b,b,g}

{g,b,g,b}

{g,b,g,g}

{g,g,b,b}

{g,g,b, g}

{b,b,g,b}

{b,b,g,g}

{b,g,b,b}

{b,g,b,g}

{b,g,g,b}

{b,g,g, g}

{b,b,b,b}

{b,b,b, g}

{g,g,g,b}

{g,g,g, g}

The letter g, b represent the girl and boy respectively.

The probability that she is the only girl from the tree diagram above= 4/16

Method 2:

The probability that the 3 children are all boys = 5/16

Then, the probability of having only 1 girl, which is the probability that the girl picked at random is the only girl in the family = 1- (the probability that the 3 children are all boys)

= 1- (5/16) = 4/16

Answer 2

1/4

Explanation:

We have a family where there are 4 children

The probability of having a girl child = 1/2

The probability of having a boy child = 1/2

Since each child has an equal chance of ending up as a boy or girl

The number of possible outcomes =

1/2 × 1/2 × 1/2 × 1/2 = 1/16

The probability of having a girl that is the only girl in this family = 4 ways

1) bbbg = 3 boys and 1 girl

2) bbgb = 2 boys, 1 girl, 1 boy

3) bgbb = 1 boy, 1 girl, 2 boys

4) gbbb = 1 girl, 3 boys

The probability to have a girl that is the only girl in the family = No of ways × No of possible outcomes

4 ways × 1/16

= 4/16 = 1/4.