Find How Many Watch All Three Games
Question:
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?
Solution
\[ n(F\cup H\cup B)=500-50 \] \[ =450 \] \[ n(F\cup H\cup B) \] \[ =n(F)+n(H)+n(B)-n(F\cap H)-n(H\cap B)-n(B\cap F)+n(F\cap H\cap B) \] \[ 450=285+195+115-70-50-45+x \] \[ 450=430+x \] \[ x=20 \]Hence, 20 viewers watch all the three games.
Exactly one game
\[ \text{Only Football}=285-70-45+20 \] \[ =190 \] \[ \text{Only Hockey}=195-70-50+20 \] \[ =95 \] \[ \text{Only Basketball}=115-45-50+20 \] \[ =40 \] \[ \text{Exactly one game}=190+95+40 \] \[ =325 \]Hence, 325 viewers watch exactly one game.