Assume that the probability of a boy being born is the same as the probability of a girl being born. Find the probability that a family with five children will have at least one boy.
1, 4/5, 13/16, 1/32, 31/32
Can someone please explain?
Thank you
at least one boy ----> exclude the case of all 5 girls, which is (1/2)^5 = 1/32
prob(at least one boy in 5) = 1 - 1/32
= 31/32
To find the probability that a family with five children will have at least one boy, we can use the concept of complement probability.
The probability of an event happening is equal to 1 minus the probability of the event not happening.
In this case, the event of interest is having at least one boy, so the complement event is not having any boys, which means having all girls.
The probability of having a girl in one birth is 1/2, and since the probability of a boy being born is the same, the probability of having a boy is also 1/2.
To find the probability of having all girls in five births, we multiply the probabilities of having a girl in each birth:
P(all girls) = P(girl) * P(girl) * P(girl) * P(girl) * P(girl) = (1/2)^5 = 1/32
Now that we have the probability of the complement event, we can find the probability of the event of interest by subtracting it from 1:
P(at least one boy) = 1 - P(all girls) = 1 - 1/32 = 31/32
Therefore, the probability that a family with five children will have at least one boy is 31/32.