Final answer:
To find the probability that fewer than four questionnaires will be responded out of 20, we can use the binomial distribution formula. The probability is approximately 0.00146.
Step-by-step explanation:
To find the probability that fewer than four questionnaires will be responded out of 20, we can use the binomial distribution formula. The formula is P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where P(X = k) is the probability of getting exactly k successes out of n trials, C(n, k) is the binomial coefficient, p is the probability of success on each trial, and n is the number of trials.
In this case, we want to find the probability that fewer than four questionnaires will be responded, so we need to calculate the probabilities of getting 0, 1, 2, and 3 responses, and add them up.
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Using the binomial distribution formula with n = 20 and p = 0.3, we can calculate the probabilities for each case:
P(X = 0) = C(20, 0) * 0.3^0 * (1-0.3)^(20-0) = 0.000000797
P(X = 1) = C(20, 1) * 0.3^1 * (1-0.3)^(20-1) = 0.000028247
P(X = 2) = C(20, 2) * 0.3^2 * (1-0.3)^(20-2) = 0.000261079
P(X = 3) = C(20, 3) * 0.3^3 * (1-0.3)^(20-3) = 0.001170207
Adding up these probabilities, P(X < 4) = 0.000000797 + 0.000028247 + 0.000261079 + 0.001170207 = 0.00146033
So, the probability that fewer than four questionnaires will be responded out of 20 is approximately 0.00146.