A survey of a group’s viewing habits over the last year revealed the following information:

1. 28% watched gymnastics
2. 29% watched baseball
3. 19% watched soccer
4. 14% watched gymnastics and baseball
5. 12% watched baseball and soccer
6. 10% watched gymnastics and soccer
7. 8% watched all three sports.

Calculate the percentage of the group that watched none of the three sports during the last year.

• 24%
• 36%
• 41%
• 52%
• 60%

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

Let

G = event that a viewer watched gymnastics

B = event that a viewer watched baseball

S = event that a viewer watched soccer

Then we want to find

[[math]] \begin{align*} \operatorname{P} [( G ∪ B ∪ S ) ] &= 1 − \operatorname{P} ( G ∪ B ∪ S ) \\ &= 1 − [ Pr ( G ) + Pr ( B ) + Pr ( S ) − Pr ( G ∩ B ) − Pr ( G ∩ S ) − Pr ( B ∩ S ) + Pr ( G ∩ B ∩ S )] \\ &= 1 − ( 0.28 + 0.29 + 0.19 − 0.14 − 0.10 − 0.12 + 0.08 ) \\ &= 1 − 0.48 \\ &= 0.52 \end{align*} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.