Binomial Distribution and its Properties

Binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials with a constant probability of success. It has two parameters: n, the number of trials and p, the probability of success in each trial.

The probability mass function of a binomial distribution is given by:

P(X=k) = nCk * p^k * (1-p)^(n-k)

where X is the number of successes, k is the number of successes, n is the number of trials, p is the probability of success in each trial, and nCk is the combination function which gives the number of ways to choose k items from n items.

Solving the Given Problem

Given, P(X=4) = 0.25P(X=2)

We know that,

P(X=k) = nCk * p^k * (1-p)^(n-k)

Substituting k=4 and k=2, we get

nC4 * p^4 * (1-p)^(n-4) = 0.25 * nC2 * p^2 * (1-p)^(n-2)

Dividing both sides by p^2 * (1-p)^(n-4), we get

nC4 * p^2 = 0.25 * nC2

nC4/nC2 = 0.25/p^2

6/5 = 0.25/p^2

p^2 = 0.25*5/6

p^2 = 0.1041667

p = sqrt(0.1041667)

p = 0.3227

Therefore, the answer is option 'D' (1/3).

Conclusion

Binomial distribution is a common probability distribution used in many fields including finance, engineering, and science. It is important to understand its properties and how to solve problems involving it. In this problem, we used the probability mass function of binomial distribution to solve for the probability of success in each trial given two probabilities of success.