A survey of 500 television viewers produced the following information : 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, 50 do not watch any of the three games. How many watch all the three games? How many watch exactly one of the three games?

By Shivank Goel|Updated : August 26th, 2022

Given the number of television viewers, n(P) = 500

Number of people who watch football, n(F) = 285

Number of people who watch hockey, n(H) = 195

Number of people who watch basketball, n(B) = 115

Number of people who watch football and basketball, n(F ∩ B) = 45

Number of people who watch football and hockey, n(F ∩ H) = 70

Number of people who watch hockey and basketball, n(H ∩ B) = 50

Number of people who do not watch all the three games, n(F ∪ H ∪ B)' = 50

Total number of viewers, n(F ∪ H ∪ B) = n(P) - n(F ∪ H ∪ B)'

= 500 - 50

n(F ∪ H ∪ B) = 450

We have to find the number of people who watch all three games.

n((F ∪ H ∪ B) = n(F) + n(H) + n(B) - n(F ∩ B) - n(F ∩ H) - n(H ∩ B) + n(F ∩ H ∩ B)

450 = 285 + 195 + 115 - 45 - 70 - 50 + n(F ∩ H ∩ B)

450 = 430 + n(F ∩ H ∩ B)

n(F ∩ H ∩ B) = 540 - 430

n(F ∩ H ∩ B) = 20

Therefore, 20 people watch all three games.

We have to find the number of people who watch exactly one of the three games.

n(exactly one game) = n(F) + n(H) + n(B) - 2n(F ∩ B) - 2n(F ∩ H) - 2n(H ∩ B) + 3n(F ∩ H ∩ B)

= 285 + 195 + 115 - 2(45) - 2(70) - 2(50) + 3(20)

= 595 - 90 - 140 - 100 + 60

= 655 - 330

= 325

Therefore, 325 people watch exactly one of the three games.

Summary:

A survey of 500 television viewers produced the following information: 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, and 50 do not watch any of the three games. How many watch all three games? How many watch exactly one of the three games?

A survey of 500 television viewers produced the following information: 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, 50 watch hockey and basketball, and 50 do not watch any of the three games. 20 people watch all three games. Three hundred twenty-five people watch exactly one of the three games.

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