Binomial Distribution:

Binomial Distribution is a discrete probability distribution that deals with the number of successes in a fixed number of independent trials [1]. It is usually denoted as B(n, p), where n is the number of trials and p is the probability of success in each trial.

Mean and Standard Deviation of Binomial Distribution:

Mean and Standard Deviation of Binomial Distribution can be calculated using the following formulas [2]:

Mean (μ) = np

Standard Deviation (σ) = √(npq)

where q = 1 - p

Solution:

Given, Mean (μ) = 20, Standard Deviation (σ) = 4

To find q, we need to use the formula of Standard Deviation of Binomial Distribution:

σ = √(npq)

Squaring both sides, we get:

σ^2 = npq

Substituting the given values, we get:

4^2 = 20q(1 - q)

16 = 20q - 20q^2

20q^2 - 20q + 16 = 0

Dividing both sides by 4, we get:

5q^2 - 5q + 4 = 0

Using the quadratic formula, we get:

q = [5 ± √(5^2 - 4*5*4)]/(2*5)

q = [5 ± √(25 - 80)]/10

q = [5 ± √(-55)]/10

As q has to be between 0 and 1, and we can't take the square root of a negative number, we discard the negative solution.

q = [5 + √(-55)]/10

q = [5 + i√55]/10

where i is the imaginary unit (√-1)

Therefore, q is equal to 4/5 (Option D).

Conclusion:

Binomial Distribution is a useful tool in probability and statistics, which helps in calculating the number of successes in a fixed number of independent trials. The mean and standard deviation of binomial distribution can be calculated using the appropriate formulas. In this question, we used the standard deviation formula to find q, and after discarding the negative solution, we arrived at the correct answer of q = 4/5.