Answer to Question #232380 in Math for kim

Question

In a survey of 232 professional athletes, it was found that 102 of them owned a convertible, 105 of them owned a giant screen TV, and 102 owned a sporting goods store. 19 owned a convertible and a store, 43 owned a TV and a store, and 56 owned a covertible and a TV. 13 owned all three items.
How many athletes did not own any of the three items?
How many owned a covertible and a TV, but not a store?
How many athletes owned a convertible or a TV?
How many athletes owned exactly one type of item in the survey?
How many athletes owned at least one type of item in the survey?
How many owned a TV or a store, but not a convertible?

Answer 1

Let $C$ denotes the event "athlete owned a convertible".

Let $T$ denotes the event "athlete owned a giant screen TV".

Let $S$ denotes the event "athlete owned a a store".

Given

$$N(U)=232, N(C)=102, N(T)=105, N(S)=102$$

$$N(C\cap T)=56, N(C\cap S)=19, N(T\cap S)=43$$

$$N(C\cap T\cap S)=13$$

Then

$$N(C\cup T\cup S)=N(C)+N(T)+N(S)$$

$$-N(C\cap T)-N(C\cap S)-N( T\cap S)$$

$$+N(C\cap T\cap S)$$

$$=102+105+102-56-19-43+13=204$$

1. How many athletes did not own any of the three items?

$$N(U)-N(C\cup T\cup S)=232-204=28$$

2. How many owned a covertible and a TV, but not a store?

$$N(C\cap T)-N(C\cap T\cap S)=56-13=43$$

3. How many athletes owned a convertible or a TV?

$$N(C\cup T)=N(C)+N(T)-N(C\cap T)$$

$$=102+105-56=151$$

4. How many athletes owned exactly one type of item in the survey?

$$204-(6+43+30+13)=112$$

5. How many athletes owned at least one type of item in the survey?

$$N(C\cup T\cup S)=204$$

6. How many owned a TV or a store, but not a convertible?

$$N(C\cup T\cup S)-N(C)=204-102=102$$