The variance will reach a maximum in a binomial distribution when:

Question

Quizplus · Accepted Answer

In a binomial distribution, the variance is given by Var(X) = np(1-p), where n is the number of trials and p is the probability of success in each trial. 
To maximize the variance, we need to maximize the product np(1-p). 
In choice A, π = 1, which means there is only one possible outcome (success) and the variance is zero. In choice D, π = 0, which also means there is only one possible outcome (failure) and the variance is zero. 
In choice B, π = ¼ and 1-π = ¾, which gives us np(1-p) = 3/16. 
In choice C, π = ½ and 1-π = ½, which gives us np(1-p) = 1/4, which is the highest value among all the choices. Therefore, choice C maximizes the variance in a binomial distribution.

The Variance Will Reach a Maximum in a Binomial Distribution