We are given that out of 120 customers, 45 bought both bags and shoes. Also, everyone bought either bags or shoes, and 11 more customers bought shoes than bags. We are asked to illustrate the information using a Venn diagram, find the number of customers who bought shoes, and calculate the probability that a customer selected at random bought bags.

1. Problem Description

We are given that out of 120 customers, 45 bought both bags and shoes. Also, everyone bought either bags or shoes, and 11 more customers bought shoes than bags. We are asked to illustrate the information using a Venn diagram, find the number of customers who bought shoes, and calculate the probability that a customer selected at random bought bags.

2. Solution Steps

(a) Venn Diagram:

Let $B$ be the number of customers who bought bags.

Let $S$ be the number of customers who bought shoes.

We are given that the total number of customers is

1

2

0. Since everyone bought either bags or shoes or both, we have:

$B \cup S = 120$

We are given that the number of customers who bought both bags and shoes is 45, so $B \cap S = 45$.

We are also given that the number of customers who bought shoes is 11 more than the number who bought bags, so $S = B + 11$.

Using the inclusion-exclusion principle, we have:

$B \cup S = B + S - B \cap S$

$120 = B + S - 45$

$120 = B + (B + 11) - 45$

$120 = 2B + 11 - 45$

$120 = 2B - 34$

$2B = 154$

$B = 77$

Then $S = B + 11 = 77 + 11 = 88$.

Now, we can calculate the number of customers who bought only bags:

$B - (B \cap S) = 77 - 45 = 32$

And the number of customers who bought only shoes:

$S - (B \cap S) = 88 - 45 = 43$

In the Venn diagram, one circle represents bags and the other represents shoes. The intersection contains 45 customers. The "bags only" section contains 32 customers, and the "shoes only" section contains 43 customers.

(b) Number of customers who bought shoes:

We already found that $S = 88$. So, 88 customers bought shoes.

(c) Probability that a customer selected at random bought bags:

The number of customers who bought bags is $B = 77$.

The total number of customers is

1

2

0. The probability that a customer selected at random bought bags is:

$P(\text{Bags}) = \frac{\text{Number of customers who bought bags}}{\text{Total number of customers}} = \frac{77}{120}$

3. Final Answer

(a) Venn Diagram: Two overlapping circles, one labeled "Bags" and the other labeled "Shoes". The intersection is

4

5. The "Bags only" part is 32, and the "Shoes only" part is

4

3. (b) The number of customers who bought shoes is

8

8. (c) The probability that a customer selected at random bought bags is $\frac{77}{120}$.