More than 200 billion grocery coupons are distributed each year for discounts exceeding $84 billion. However, according to a report in USA Today, only 3.2% of the coupons are redeemed. If a company distributes 5000 coupons, what is the probability that:
(a) at least 100 coupons are redeemed?
(b) at most 200 coupons are redeemed?
(c) fewer than 100 coupons are not redeemed?
(d) more than 200 coupons are not redeemed?
To solve these probability questions, we can use the binomial distribution formula. The binomial distribution is used when we have a fixed number of independent trials with two possible outcomes, success or failure.
Let's break down each question and explain how to calculate the probabilities:
(a) At least 100 coupons are redeemed:
To find the probability that at least 100 coupons are redeemed, we need to find the sum of probabilities for 100 or more redeemed coupons.
1) Calculate the probability of redeeming exactly 100 coupons:
- p = 0.032 (given in the question)
- n = 5000 (the total number of coupons distributed)
- P(X = 100) = C(n, X) * p^X * (1-p)^(n-X)
= C(5000, 100) * 0.032^100 * (1-0.032)^(5000-100)
2) Calculate the probability of redeeming more than 100 coupons:
- P(X > 100) = P(X = 101) + P(X = 102) + ... + P(X = 5000)
- P(X > 100) = C(5000, 101) * 0.032^101 * (1-0.032)^(5000-101) + ...
(b) At most 200 coupons are redeemed:
To find the probability that at most 200 coupons are redeemed, we need to find the sum of probabilities for 200 or less redeemed coupons.
1) Calculate the probability of redeeming exactly 200 coupons:
- P(X = 200) = C(5000, 200) * 0.032^200 * (1-0.032)^(5000-200)
2) Calculate the probability of redeeming less than 200 coupons:
- P(X < 200) = P(X = 0) + P(X = 1) + ... + P(X = 199)
- P(X < 200) = C(5000, 0) * 0.032^0 * (1-0.032)^(5000-0) + ...
(c) Fewer than 100 coupons are not redeemed:
To find the probability that fewer than 100 coupons are not redeemed, we need to find the sum of probabilities for 0 to 99 coupons not being redeemed.
1) Calculate the probability of exactly 99 coupons not being redeemed:
- P(X = 99) = C(5000, 99) * (1-0.032)^99 * 0.032^(5000-99)
2) Calculate the probability of fewer than 100 coupons not being redeemed:
- P(X < 100) = P(X = 0) + P(X = 1) + ... + P(X = 99)
- P(X < 100) = C(5000, 0) * (1-0.032)^0 * 0.032^(5000-0) + ...
(d) More than 200 coupons are not redeemed:
To find the probability that more than 200 coupons are not redeemed, we need to find the sum of probabilities for 200 or less coupons being redeemed.
1) Calculate the probability of exactly 200 coupons being redeemed:
- P(X = 200) = C(5000, 200) * 0.032^200 * (1-0.032)^(5000-200)
2) Calculate the probability of more than 200 coupons not being redeemed:
- P(X > 200) = P(X = 201) + P(X = 202) + ... + P(X = 5000)
- P(X > 200) = C(5000, 201) * (1-0.032)^201 * 0.032^(5000-201) + ...
Remember to substitute the values of C(n, X) using the combination formula: C(n, X) = n! / (X!(n-X)!), where n! denotes the factorial of n.
Once you have all the required probabilities, you can calculate them using a calculator or software that supports binomial probability calculations.