Question

The Mean and Variance of a Binomial distribution are 4 and 4/3 respectively. Find the probability of

i) Two successes

ii) More than Two successes

iii) Three or more than three successes​

Answer 1

The probabilities, we use the mean and variance formulas of a binomial distribution. We calculate the probability of two successes, more than two successes, and three or more successes using the probability mass function (PMF) of the binomial distribution.

Step-by-step explanation:

To find the probability, we can use the formulas for the mean and variance of a binomial distribution. The mean (µ) is equal to np, and the variance (σ^2) is equal to npq, where n is the number of trials, p is the probability of success, and q is the probability of failure.

i) To find the probability of two successes, we can use the probability mass function (PMF) of the binomial distribution. The PMF is given by P(X = x) = C(n, x) * p^x * q^(n-x), where C(n, x) is the number of combinations of n items taken x at a time. Plugging in the values, we have P(X = 2) = C(n, 2) * p^2 * q^(n-2).

ii) To find the probability of more than two successes, we can sum up the probabilities of three or more successes. P(X > 2) = P(X = 3) + P(X = 4) + .... This can be calculated using the PMF.

iii) To find the probability of three or more than three successes, we can sum up the probabilities of three or more successes. P(X ≥ 3) = P(X = 3) + P(X = 4) + ... This can also be calculated using the PMF.