Historically, 5 percent of a mail-order firm’s repeat charge-account customers have an incorrect current address in the firm’s computer database. (a) What is the probability that none of the next 12 repeat customers who call will have an incorrect address? (b) One customer? (c) two customers
To calculate the probability, we need to use the binomial probability formula, which is given by:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials
x is the number of successful trials
p is the probability of success in a single trial
Now let's solve the problem step-by-step:
(a) Probability that none of the next 12 customers will have an incorrect address:
In this case, x = 0 (no customers have an incorrect address), n = 12 (total number of customers), and p = 0.05 (probability of having an incorrect address).
P(0) = (12C0) * 0.05^0 * (1-0.05)^(12-0)
= 1 * 1 * 0.95^12
≈ 0.5613
Therefore, the probability that none of the next 12 customers will have an incorrect address is approximately 0.5613, or 56.13%.
(b) Probability that one customer will have an incorrect address:
In this case, x = 1 (one customer has an incorrect address), n = 12, and p = 0.05.
P(1) = (12C1) * 0.05^1 * (1-0.05)^(12-1)
= 12 * 0.05 * 0.95^11
≈ 0.3095
Therefore, the probability that one customer will have an incorrect address is approximately 0.3095, or 30.95%.
(c) Probability that two customers will have an incorrect address:
In this case, x = 2, n = 12, and p = 0.05.
P(2) = (12C2) * 0.05^2 * (1-0.05)^(12-2)
= 66 * 0.05^2 * 0.95^10
≈ 0.0746
Therefore, the probability that two customers will have an incorrect address is approximately 0.0746, or 7.46%.