According to a recent survey, 65% of all customers will return to the same grocery store. Suppose 11 customers are selected at random,what is the probability that
A) exactly 5 will return?
B) all 11 will return?
C) at least 6 will return
D) at least one will return
E) how many customers would be expected to return to the same store
Please explain I want to understand how to do it :) thanks in advance!
To find probabilities in this scenario, we can use the binomial probability formula:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting k successes.
- n is the total number of trials (in this case, the total number of customers selected).
- k is the number of desired successes.
- p is the probability of success in a single trial (in this case, the probability of a customer returning).
- (nCk) is the binomial coefficient, calculated as n! / (k! * (n-k)!).
Let's solve each part of the question step-by-step:
A) The probability that exactly 5 customers will return:
In this case, n = 11 (since 11 customers are selected) and p = 0.65 (as given in the survey). We need to calculate P(X=5).
Using the binomial probability formula:
P(X=5) = (11C5) * (0.65^5) * (1-0.65)^(11-5)
= (11! / (5! * (11-5)!)) * (0.65^5) * (0.35^6)
Calculate the numerator, denominator, and exponent parts separately:
Numerator: 11! / (5! * (11-5)!) = 11! / (5! * 6!) = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1) = 462.
Now substitute the values:
P(X=5) = 462 * (0.65^5) * (0.35^6)
Use a calculator to evaluate this expression for the final answer.
B) The probability that all 11 customers will return:
In this case, we need to calculate P(X=11), where n = 11 and p = 0.65.
Using the binomial probability formula as before:
P(X=11) = (11C11) * (0.65^11) * (1-0.65)^(11-11)
= (11! / (11! * (11-11)!)) * (0.65^11) * (0.35^0)
= 0.65^11
Evaluate this expression using a calculator.
C) The probability that at least 6 customers will return:
To find this probability, we need to calculate the probabilities of 6, 7, 8, 9, 10, and 11 customers returning, and sum them up.
P(X>=6) = P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) + P(X=11)
Use the binomial probability formula to find each individual probability and then add them up.
D) The probability that at least one customer will return:
This probability can be found as the complement of the probability that no customer returns.
P(at least one) = 1 - P(none)
P(none) = P(X=0) = (11C0) * (0.65^0) * (0.35^11)
Evaluate this expression for P(none) and subtract it from 1 to find P(at least one).
E) The expected number of customers returning to the same store:
The expected value can be calculated by multiplying the total number of customers (n) by the probability of success (p).
Expected value = n * p = 11 * 0.65
Evaluate this expression to find the expected number of customers returning.