In a Binomial distribution, the mean is two times its variance. What is the probability of exactly 2 successes out of 4 trials?

Concept:

Binomial Formula. Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is:

b(X; n, P) = ⁿC_r . P^x . (1 - P)^{n - x}

For a binomial distribution, the mean, variance and standard deviation for the given number of success are represented using the formulas

Mean μ = np

Variance σ2 = np(1 - p) = npq

Where p is the probability of success

q is the probability of failure, where q = 1 - p

Calculation:

Given: The mean is two times its variance

Mean = 2 × Variance

np = 2 × npq

q = \(\rm \frac{1}{2}\)

p = 1 - q = 1 - \(\rm \frac{1}{2}\) = \(\rm \frac{1}{2}\)

Given: n = 4 trials and r = 2

We know P (x = 2) = ⁿC_r. p^r. q^{n - r}

= 4C2. \(\rm (\frac{1}{2})\)2. \(\rm (\frac{1}{2})\)4 - 2

= 4C2. \(\rm (\frac{1}{2})\)2. \(\rm (\frac{1}{2})\)2

= 4C2. \(\rm \frac{1}{4}\). \(\rm \frac{1}{4}\)

= 6 × \(\rm \frac{1}{4}\) × \(\rm \frac{1}{4}\)

= \(\rm \frac{3}{8}\)

Additional Information

Properties of Binomial Distribution

The properties of the binomial distribution are:

• There are two possible outcomes: true or false, success or failure, yes or no.
• There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
• The probability of success or failure varies for each trial.
• Only the number of successes is calculated out of n independent trials.
• Every trial is an independent trial, which means the outcome of one trial does not affect the outcome of another trial.