# What is the probability of havingat least one boy out of 3 children?Please show work.

sociality | High School Teacher | (Level 1) Valedictorian

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Every child born can be either a boy or a girl. So we have two options here. Therefore the probability of a boy being born is 1/2.

Now for each of the three children that are born there is a 1/2 probability of a boy being born. And the probability of a girl being born is also 1/2.

The probability that at least one of the children being born is a boy is equal to 1- ( probability that all three are born girls) = 1 - [ (1/2)*(1/2)*(1/2)] = 1- 1/8 = 7/8.

The required probability of having at least one boy out of three children is 7/8.

neela | High School Teacher | (Level 3) Valedictorian

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We assume the a child is either a boy or a girl. Therefore the probability that a child is a boy = 1/2. Also the probability that child is a girl  = 1/2

Therefore the event of at least one boy out 3 children includes the event of one boy and 2 girls,  2 boys and 1 girl, all 3 boys .

The probability that one child is a boy and 2 are girls = 3C1*(1/2) {1-1/2}^2 = 3/8.

The probability that 2 are boys and one is girl = 3C2*(1/2)^2 {1-1/2} =3/8

The pobability that all 3 are boys = 3C3*(1/2)^3 = 1/8

Therefore the required probability  of at leat least one child is boy out of 3 children = 3/8+3/8+1/8 = 7/8.

Alternatively , probability of at least one out of 3 chidren is a boy is equivalent to the probability of not all 3 are girls. Probability that all three children are girls = (1/2)^3 = 1/8. Therefore the probability that not all three children are girls is 1-1/8 = 7/8.