Problem 9 - A First Course in Probability

Question

A retail establishment accepts either the American Express or the VISA credit card. A total of 24 percent of its customers carry an American Express card, 61 percent carry a VISA card, and 11 percent carry both cards. What percentage of its customers carry a credit card that the establishment will accept?

Step-by-Step Solution

Answer

74% of the customers carry a credit card that the retail establishment will accept, using the principle of inclusion and exclusion: |A ∪ V| = |A| + |V| - |A ∩ V|, with |A| = 24, |V| = 61, and |A ∩ V| = 11.

Step 1: Understanding the principle of inclusion and exclusion

The principle of inclusion and exclusion is a counting method used in combinatorics to find the number of elements in the union of multiple sets. In this exercise, we have two sets: the set of customers carrying an American Express card (A) and the set of customers carrying a VISA card (V). We want to find the percentage of customers who carry either an American Express or a VISA card (which is the union of these two sets). The principle of inclusion and exclusion states that: |A ∪ V| = |A| + |V| - |A ∩ V| where |A ∪ V| is the number of elements in the union of sets A and V, |A| is the number of elements in set A, |V| is the number of elements in set V, and |A ∩ V| is the number of elements in the intersection of sets A and V (i.e., customers carrying both cards).

Step 2: Assigning percentages to each set

We are given the following percentages: - 24% of customers carry an American Express card, so |A| = 24. - 61% of customers carry a VISA card, so |V| = 61. - 11% of customers carry both cards, so |A ∩ V| = 11.

Step 3: Apply the principle of inclusion and exclusion

Using the principle of inclusion and exclusion, we can find the percentage of customers who carry at least one of the accepted credit cards: |A ∪ V| = |A| + |V| - |A ∩ V| Substitute the given values: |A ∪ V| = 24 + 61 - 11

Step 4: Calculate the percentage of customers carrying an accepted card

Now, calculate the sum: |A ∪ V| = 13 + 61 = 74 Therefore, 74% of the customers carry a credit card that the retail establishment will accept.