Question

in a survey of 211 professional athletes, it was found that 99 of them owned a convertible, 85 of them owned a giant screen tv, and 92 owned a sporting goods store. 23 owned a convertible and a store, 31 owned a tv and a store, and 46 owned a convertible and a tv. 8 owned all three items. 1. how many athletes did not own any of the three items? 2. how many owned a convertible and a tv, but not a store? 3. how many athletes owned a convertible or a tv? 4. how many athletes owned exactly one type of item in the survey? 5. how many athletes owned at least one type of item in the survey? 6. how many owned a tv or a store, but not a convertible?

Explanation:

Step1: Recall the principle of inclusion - exclusion formula

Let \(A\) be the set of athletes who own a convertible, \(|A| = 99\), \(B\) be the set of athletes who own a giant - screen TV, \(|B|=85\), and \(C\) be the set of athletes who own a sporting goods store, \(|C| = 92\). Also, \(|A\cap C|=23\), \(|B\cap C| = 31\), \(|A\cap B|=46\), and \(|A\cap B\cap C| = 8\). The formula for \(|A\cup B\cup C|\) is \(|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|\).

Step2: Calculate \(|A\cup B\cup C|\)

\[

$$\begin{align*} |A\cup B\cup C|&=99 + 85+92-46 - 23-31 + 8\\ &=(99 + 85+92)+8-(46 + 23+31)\\ &=276+8 - 100\\ &=184 \end{align*}$$

\]

Step3: Answer question 1

The number of athletes who did not own any of the three items is the total number of athletes minus \(|A\cup B\cup C|\). The total number of athletes is \(n = 211\). So the number is \(211-184=27\).

Step4: Answer question 2

The number of athletes who owned a convertible and a TV but not a store is \(|A\cap B|-|A\cap B\cap C|=46 - 8=38\).

Step5: Answer question 3

The number of athletes who owned a convertible or a TV is \(|A\cup B|=|A|+|B|-|A\cap B|=99 + 85-46=138\).

Step6: Answer question 4

The number of athletes who own only a convertible is \(|A|-(|A\cap B|+|A\cap C|)+|A\cap B\cap C|=99-(46 + 23)+8=38\).
The number of athletes who own only a TV is \(|B|-(|A\cap B|+|B\cap C|)+|A\cap B\cap C|=85-(46 + 31)+8=16\).
The number of athletes who own only a store is \(|C|-(|A\cap C|+|B\cap C|)+|A\cap B\cap C|=92-(23 + 31)+8=46\).
The number of athletes who own exactly one type of item is \(38 + 16+46=100\).

Step7: Answer question 5

The number of athletes who owned at least one type of item is \(|A\cup B\cup C| = 184\).

Step8: Answer question 6

The number of athletes who own a TV or a store but not a convertible:
First, \(|(B\cup C)\cap\overline{A}|=|B\cup C|-|(B\cup C)\cap A|\).
We know \(|B\cup C|=|B|+|C|-|B\cap C|=85 + 92-31=146\).
And \(|(B\cup C)\cap A|=(|A\cap B|+|A\cap C|)-|A\cap B\cap C|=(46 + 23)-8=61\).
So \(|(B\cup C)\cap\overline{A}|=146-61 = 85\).

Answer:

1. 27
2. 38
3. 138
4. 100
5. 184
6. 85