A MAIL-ORDER FIRM HAS A CIRCULAR WHICH ELICITS A 10 % RESPONSE RATE. SUPPOSE 20 OF THE CIRCULARS ARE MAILED AS A MARKET TEST IN A NEW GEOGRAPHIC AREA. ASSUMING THAT THE 10% RESPONSE RATE IS APPLICABLE IN THE NEW AREA, DETERMINE THE PROBABILITIES OF THE FOLLOWING EVENTS:
A. NO ONE RESPONDS
B. A MAJORITY OF THE PEOPLE RESPOND
C. WHAT IS THE EXPECTED NUMBER OF PEOPLE WILL RESPOND?
To determine the probabilities of the given events, we need to use the concept of binomial distribution. In this scenario, we have a 10% response rate, meaning each circular has a 10% chance of receiving a response.
The binomial distribution formula is given by P(x) = (nCx) * p^x * (1-p)^(n-x), where:
- P(x) is the probability of getting x successes,
- n is the total number of trials,
- p is the probability of success in a single trial, and
- (nCx) is the binomial coefficient, representing the number of ways to choose x successes from n trials.
Let's calculate the probabilities of the given events:
A. No one responds:
To find the probability that no one responds, we need to calculate P(x=0).
- n = 20 (number of circulars mailed)
- p = 0.1 (response rate)
Using the binomial distribution formula:
P(x=0) = (20C0) * 0.1^0 * (1-0.1)^(20-0)
= 1 * 1 * 0.9^20
≈ 0.1216
Therefore, the probability of no one responding is approximately 0.1216, or approximately 12.16%.
B. Majority of the people respond:
To find the probability that a majority of the people respond, we need to calculate P(x > 10).
In this case, half the number of circulars plus one (11 or more) responding will be considered a majority.
To calculate the cumulative probability, we need to sum P(x) for x=11, 12, 13, ..., 20.
- n = 20
- p = 0.1
P(x>10) = P(x=11) + P(x=12) + ... + P(x=20)
Using the binomial distribution formula:
P(x>10) = Σ (20Ck) * 0.1^k * 0.9^(20-k) for k=11 to 20
By calculating these individual probabilities and summing them, we find that P(x>10) ≈ 0.7471.
Therefore, the probability of a majority of people responding is approximately 0.7471, or approximately 74.71%.
C. Expected number of people who will respond:
The expected number of people who will respond (E(x)) is calculated as the product of the total number of trials and the probability of success in a single trial.
- n = 20
- p = 0.1
E(x) = n * p
= 20 * 0.1
= 2
Therefore, the expected number of people who will respond is 2.