A shoe manufacturer wants to find out which type of shoe is the most popular. They ask 100 people which of type $A$, type $B$ or type $C$ they use. Of these people, 14 use all three types 10 do not use any of type $A$, type $B$ or type $C$ 23 use both type $A$ and type $B$, but not type $C$ 34 use both type $A$ and type $C$ 56 in total use type $B$ all 40 who use type $C$ also use at least one of type $A$ and type $B$. a) Complete the Venn diagram for this information. (4) One of the customers is chosen at random. Given that this customer uses type $B$, b) find the probability that this customer also uses type $A$. (5) Total marks: 6

Step 1 ：Let A, B, and C be the sets of people who use type A, type B, and type C shoes, respectively. We are given the following information:

Step 2 ：1. 14 people use all three types of shoes.

Step 3 ：2. 10 people do not use any type of shoe.

Step 4 ：3. 23 people use both type A and type B, but not type C.

Step 5 ：4. 34 people use both type A and type C.

Step 6 ：5. 56 people in total use type B.

Step 7 ：6. All 40 people who use type C also use at least one of type A and type B.

Step 8 ：Using this information, we can find the number of people who use each type of shoe:

Step 9 ：Total people who use type A: \(57\)

Step 10 ：Total people who use type B: \(56\)

Step 11 ：Total people who use type C: \(40\)

Step 12 ：People who use type A and type B, but not type C: \(23\)

Step 13 ：People who use type A and type C, but not type B: \(20\)

Step 14 ：People who use type B and type C, but not type A: \(26\)

Step 15 ：People who use all three types: \(14\)

Step 16 ：People who do not use any type: \(10\)

Step 17 ：Complete the Venn diagram with this information.

Step 18 ：Given that a customer uses type B, the probability that this customer also uses type A is approximately \(\boxed{0.661}\) (rounded to three decimal places).