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 need to use the concept of the binomial distribution. In the case of coupon redemption, we can consider each coupon as a Bernoulli trial, where it has a certain probability of being redeemed or not redeemed.
The binomial distribution is governed by two parameters: the number of trials (n) and the probability of success (p). In this case, the number of trials is 5000 (since the company distributed 5000 coupons), and the probability of success is 0.032 (3.2% redemption rate).
To calculate these probabilities, we will use the binomial probability formula, which is:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- P(X=k) is the probability of exactly k successes
- n C k is the number of combinations of n items taken k at a time (calculated as n! / (k! * (n-k)!))
- p^k is the probability of k successes
- (1-p)^(n-k) is the probability of n-k failures
Now, let's solve each part of the question:
(a) At least 100 coupons are redeemed:
To find this probability, we need to sum the probabilities of 100 or more coupons being redeemed. We can use the complement rule to calculate the probability of fewer than 100 coupons being redeemed and subtract it from 1.
P(X >= 100) = 1 - P(X < 100)
(b) At most 200 coupons are redeemed:
To find this probability, we need to sum the probabilities of 200 or fewer coupons being redeemed.
P(X <= 200) = P(X = 0) + P(X = 1) + ... + P(X = 200)
(c) Fewer than 100 coupons are not redeemed:
To find this probability, we need to sum the probabilities of 0 to 99 coupons not being redeemed.
P(X < 100) = P(X = 0) + P(X = 1) + ... + P(X = 99)
(d) More than 200 coupons are not redeemed:
To find this probability, we need to sum the probabilities of 201 or more coupons not being redeemed.
P(X > 200) = P(X = 201) + P(X = 202) + ... + P(X = n)
By plugging in the values of n, k, and p into the binomial probability formula, you can calculate the probabilities for each part of the question.