Question

The Federal Deposit Insurance Corporation is a United States government corporation providing deposit insurance to depositors in U.S. commercial banks and savings institutions. The FDIC was created by the 1933 Banking Act, enacted during the Great Depression to restore trust in the American banking system. The FDIC recently conducted a survey and found that 45% of all financial consumers were very satisfied with their primary financial institution. If this figure still holds true today, suppose 28 financial consumers are sampled randomly. If X is a binomial random variable representing the number of financial consumers who were very satisfied with their primary financial institution, what is the probability that exactly 20 of the 28 are very satisfied with the primary financial institution? (Report your answer to 4 decimal places, using conventional rounding rules.)

Answer 1

Let X be a number of customer satisfied with the primary financial instistuation among 28 customers.

Here, X ~ Binomial ( n = 28, p = 0.45)

Probability mass function of X is,

P(X = x) = ⁿC_x p^x (1 - p)^n-x

We want to find, P(X = 20)

P(X = 20)

= ²⁸C₂₀ * (0.45)²⁰ * (1 - 0.45)^28-20

= 3108105 * (0.45)²⁰ * (0.55)⁸

= 0.00301750

=> P(X = 20) = 0.0030 (rounded to 4 decimal places)

Therefore, required probability is 0.0030

Note: Using Excel we can find this probability by,

P(X = 20) = BINOMDIST(20, 28, 0.45, 0) = 0.00301750