To solve this, we can use a Venn diagram or a table. Let's create a table to keep track of the information given:
Football Hockey Basketball Total
------------------------------------
Yes 285 195 115 ?
Football
and
Basketball 45 ? 50 ?
Football
and
Hockey 70 ? ? ?
Hockey
and
Basketball ? 50 ? ?
No ? ? ? 50
------------------------------------
To find the missing values in the table, we can use the following formulas:
Total = Football + Hockey + Basketball - (Football and Basketball) - (Football and Hockey) - (Hockey and Basketball) + (Football, Hockey, and Basketball)
or
Total = (Football only) + (Hockey only) + (Basketball only) + (Football and Hockey and Basketball) + (None of the three games)
Now let's fill in the table:
Football Hockey Basketball Total
------------------------------------
Yes 285 195 115 ?
Football
and
Basketball 45 ? 50 ?
Football
and
Hockey 70 ? ? ?
Hockey
and
Basketball ? 50 ? ?
No ? ? ? 50
------------------------------------
From the given information, we know that 50 people do not watch any of the three games. Let's write that in the table:
Football Hockey Basketball Total
------------------------------------
Yes 285 195 115 ?
Football
and
Basketball 45 ? 50 ?
Football
and
Hockey 70 ? ? ?
Hockey
and
Basketball ? 50 ? ?
No ? ? ? 50
------------------------------------
50
To find the total number of viewers, we can use the formula:
Total = (Football only) + (Hockey only) + (Basketball only) + (Football and Hockey and Basketball) + (None of the three games)
Plugging in the given values, we have:
Total = (285) + (195) + (115) + (45) + (70) + (50) + (50) = 810
Now let's fill in the remaining missing values:
Football Hockey Basketball Total
------------------------------------
Yes 285 195 115 ?
Football
and
Basketball 45 20 50 ?
Football
and
Hockey 70 5 10 ?
Hockey
and
Basketball 25 50 15 ?
No 150 80 20 50
------------------------------------
810
1) How many watch all three games?
From the table, we see that the value in the Football and Hockey and Basketball intersection is 10. So, 10 viewers watch all three games.
2) How many watch exactly one of the three games?
To find the value, we sum up the values in the Football only, Hockey only, and Basketball only columns:
Football only: 150
Hockey only: 80
Basketball only: 20
Total: 150 + 80 + 20 = 250
So, 250 viewers watch exactly one of the three games.