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 binomial distribution formula. The binomial distribution is a discrete probability distribution that measures the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success.
In this case, we can use the binomial distribution formula because each coupon redemption can be considered an independent Bernoulli trial, and the probability of success (redeeming a coupon) remains the same for each trial.
The formula for the probability of x successes in n trials with a probability of success p is:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
Where:
P(x) is the probability of x successes
nCx is the number of combinations of n items taken x at a time (calculated as n! / (x! * (n-x)!)
p is the probability of success
x is the number of successes
n is the total number of trials
Now let's solve each part of the problem:
(a) At least 100 coupons are redeemed:
To find the probability of at least 100 coupons being redeemed, we need to calculate the probabilities of 100, 101, 102, ... up to the maximum possible number of redemptions. In this case, the maximum number of redemptions is 5000 (the total number of coupons distributed).
P(x >= 100) = P(x=100) + P(x=101) + ... + P(x=5000)
To calculate this probability, we need to use the binomial distribution formula for each value of x and sum them up.
(b) At most 200 coupons are redeemed:
To find the probability of at most 200 coupons being redeemed, we need to calculate the probabilities of 0, 1, 2, ... up to 200.
P(x <= 200) = P(x=0) + P(x=1) + ... + P(x=200)
(c) Fewer than 100 coupons are not redeemed:
To find the probability of fewer than 100 coupons not being redeemed, we need to calculate the probabilities of 0, 1, 2, ... up to 99.
P(x < 100) = P(x=0) + P(x=1) + ... + P(x=99)
(d) More than 200 coupons are not redeemed:
To find the probability of more than 200 coupons not being redeemed, we need to calculate the probabilities of 201, 202, ... up to 5000.
P(x > 200) = P(x=201) + P(x=202) + ... + P(x=5000)
To calculate these probabilities, you will need to use a statistical software, programming language, or a binomial distribution calculator.