Customers at a gas station pay with a credit card (A), debit card (B), or cash (C). Assume that successive customers make independent choices with P (A) = 0.6, P (B) = 0.3, and P (C) = 0.1.

(a) Among the next 100 customers, what are the mean and variance of the number who pay with a debit card?

Mean = 20

Variance = 16

Step-by-step explanation:

Solution:-

- Let X be the number of customers paying with a debit card. X has the binomial distribution with n = 100 trials and success probability p = 0.2

- In general, if X has the binomial distribution with (n) trials and a success probability of (p) then:

P[X = x] = n! / (x! (n-x) !) * p^x * (1-p) ^ (n-x)

for values of x = 0, 1, 2, ..., n

P[X = x] = 0 for any other value of x.

- The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.

- Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.

X ~ Binomial (n = 100, p = 0.2)

- The mean of the binomial distribution is n * p = 20

- The variance of the binomial distribution is n * p * (1 - p) = 16