Question

What is the probability that out of 175 chicks hatched on Peeper Farm, at least 80 will be female? Assume that males and females are equally probable, and round your answer to the nearest tenth of a percent.

Answer 1

Since we have two possible outcomes for each hatch and they are independent of each other this can be seen as a binomial experiment.

We know that the binomial distribution is given by:

`P(X=k)=(n!)/(k!(n-k)!)p^k(1-p)^(n-k)`

where n is the number of trials, k is the number of succeses we want and p is the probability of success.

In this case we have 175 chicks hatched, then the number of trials is n=175; the probability of success is 0.5 (since the probability of having males or females is equal) and we want that the number of succeses to be at least 80, then we are looking for the probability:

`P(X\ge80)`

but using the properties of the distribution this is the same as:

`P(X\ge80)=P(X=80)+P(X=81)+P(X=82)+\cdot\cdot\cdot+P(X=175)`

where each probability is calculated with the values given above.

In this case we would have to calculate 95 probabilities, as an example we would calculate the first one:

`P(X=80)=(175!)/(80!(175-80)!)(0.5)^(80)(1-0.5)^(175-80)=0.03175876249`

By doing all the probabilities and adding them we have that the probability is 0.887 that's the same as 88.7%, therefore the answer is b.