Binomial distribution is the discrete probability distribution with parameters n and p. This is the basis for the popular binomial test of statistical significance. In a binomial distribution there are always two mutually exclusive events (as binomial means two).

They are termed as fail and success. It is very much clear that the sum of fail and success probabilities will be 1. This further indicates that if the probability for fail is ‘f’, then the probability for success will be 1 - f.

They are termed as fail and success. It is very much clear that the sum of fail and success probabilities will be 1. This further indicates that if the probability for fail is ‘f’, then the probability for success will be 1 - f.

The formula that we commonly used for determining the distribution
of binomial types is more generally known also as the formula for
“Bernoulli trials” (when n = 1).

Let us assume that in an
experiment done, ‘n’ is representing the number of trials attempted, and
that ‘k’ is the count of successes that is to be attained in those ‘n’
trials. This implies that number of failures clearly will be ‘n - k’.

Assuming, ‘s’ to be the probability of succeeding in a trial, we get that the probability of failure is ‘1 - s’.

Then the formula for calculating the achievement of ‘k’ successes in ‘n’ trials is given below:

P (‘k’ successes in ‘n’ trials) = $C (n, k) s^k (1 - s)^(n - k)$

C
(n, k) is called the coefficient for binomial distribution or binomial
coefficient. It is on this coefficient that the distribution is named.

The
factorial of any number ‘m’ is the product of all natural numbers
starting from m, (m - 1) to 1.

C (n, k) = $\frac{n!}{(k! (n - k)! )}$

With n! = n * (n - 1) * … * 2 * 1

If k > $\frac{n}{2}$ then the following is applicable,

f (k, n, s) = f (n - k, n, 1 - s)

This holds true for every k > $\frac{n}{2}$.

Here, n = 20, n - k = 5, k = 20 - 5 = 15

Here the probability of success = probability of giving a right answer = s = $\frac{1}{4}$

Hence, the probability of failure = probability of giving a wrong answer = 1 - s

= 1 - $\frac{1}{4}$ = $\frac{3}{4}$

When we substitute these values in the formula for Binomial distribution we get,

So, P (exactly 5 out of 20 answers incorrect) = C (20, 5) * $(\frac{1}{4})$ $^{15}$ * $(\frac{3}{4})$ $^5$

$\rightarrow$ P (5 out of 20) = $\frac{(20 * 19 * 18 * 17 * 16)} {(5 * 4 * 3 * 2 * 1)}$ * $(\frac{1}{4})$ $^15$ * $(\frac{3}{4})$ $^5$

= 0.0000034 (approximately)

Thus the required probability is 0.0000034 approximately.

Here, n = 50, k = 5, n - k = 45.

The probability of success = probability of getting a “D”= s = $\frac{1}{5}$

Hence, the probability of failure = probability of not getting a “D” = 1 - s = $\frac{4}{5}$.