Im having problems with a question:
A survey of 500 television watchers produced the following information:
285 watch football games
195 watch hockey games
115 watch basketball games
45 watch both football and basketball games
70 watch football and hockey games
50 watch hockey and basketball games
50 do not watch any of the three kinds of games
A) How many people in the survey watch all three kinds of games?
B) How many people watch exactly one of the sports?
USING VENN DIAGRAM ONLY
Answers
Let F, H, B be the sets of the students who watch football, hockey, basketball respectively,
Now given data is as follows:
n(F) = 285
n(H) = 195
n(B) = 115
n(F∩B) = 45
n(F∩H) = 70
n(H∩B) = 50
Since, 50 do no watch any kind of three games, So we can remove then in our counting.
n(H∪B∪F) = 500 - 50 = 450
Number of people who watch all these games, will be intersection of set H,B and F.
We know that,
n(H∪B∪F)= n(H) + n(B) + n(F) - n(H∩B) - n(B∩F) - n(H∩F) - n(H∩B∩F)
450 = 285 + 195 + 115 - 50 - 45 - 70 - n(H∩B∩F)
450 = 595 - 165 - n(H∩B∩F)
450 = 430 - n(H∩B∩F)
n(H∩B∩F) = 20
Number of people who only watch football
n(F - B - H) = n(F) - n(F∩H) - n(F∩B) + n(H∩B∩F)
= 285 - 70 - 45 + 20
= 190
Number of people who only watch hockey
n(H - F - B) = n(H) - n(H∩F) - n(H∩B) + n(H∩B∩F)
= 195 - 70 - 50 + 20
= 95
Number of people who only watch basketball
n(B - F - H) = n(B) - n(B∩H) - n(B∩F) + n(H∩B∩F)
= 115 - 50 - 45 + 20
= 40
Hence,
Number of people who excatly watch one of the sports = 40 + 190+ 95
= 325