A store owner decides to put a coupon in the Sunday newspaper. The advertising firm she works believes that at least 20% but no more than 30% of people who shop at his store will use the coupon. The paper circulates 50,000 copies of the Sunday paper and ALL her customers get the Sunday paper. She believes that everyone shopping at her store receives the Sunday paper. What is the probability that 50% or more of the customers in a single day will have a coupon?
To find the probability that 50% or more of the customers in a single day will have a coupon, we need to consider the range of percentages given (20% to 30%) and the total number of customers (50,000).
First, let's calculate the minimum and maximum number of customers who will use the coupon based on the given range.
Minimum:
20% of 50,000 = 0.20 * 50,000 = 10,000 customers
Maximum:
30% of 50,000 = 0.30 * 50,000 = 15,000 customers
Now, let's calculate the probability of having at least 10,000 customers or more:
P(X ≥ 10,000) = 1 - P(X < 10,000)
To find the probability of having fewer than 10,000 customers using the coupon, we can calculate the cumulative probability of having exactly 9,999 customers (or fewer), using the binomial probability formula:
P(X < 10,000) = Σ [P(X = k)]
for k = 0 to 9,999
Using the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where:
n = total number of customers = 50,000
k = number of customers using the coupon (from 0 to 9,999)
p = probability of a customer using the coupon (between 0.20 and 0.30)
Finally, we can substitute the values into the formula and calculate the cumulative probability.
P(X < 10,000) = Σ [C(50,000, k) * p^k * (1-p)^(50,000-k)]
for k = 0 to 9,999
After calculating the cumulative probability, subtract it from 1 to get the probability of having at least 10,000 customers or more:
P(X ≥ 10,000) = 1 - P(X < 10,000)
This will give you the probability that 50% or more of the customers in a single day will have a coupon.