In a survey of 500 families produces the following information 285 watch football 195 watch Cookie 115 basketball football and basketball 70 watch football and hockey and basketball 50 watch hockey and basketball if 50 donot watch any of the three. how many watch all the twice.
To find out how many families watch all three sports, you need to use the principle of inclusion-exclusion.
First, add up the number of families watching each sport individually:
285 watch football
195 watch cricket
115 watch basketball
Next, subtract the number of families watching pairs of sports from the total of each sport:
70 watch football and basketball
50 watch football and hockey and basketball
50 do not watch any of the three
Now, let's calculate the number of families watching all three sports:
Total families watching football = 285
Total families watching basketball = 115
Total families watching hockey = 50
To find the number of families watching all three sports, subtract:
285 - (70 + 50) = 285 - 120 = 165
Therefore, 165 families watch all three sports.
To find the number of families that watch all three sports, we need to use the principle of inclusion-exclusion.
Let's break down the information given into a Venn diagram:
Let A = Families that watch football
Let B = Families that watch basketball
Let C = Families that watch hockey
From the given information:
A = 285
B = 195
C = 115
A ∩ B = 115
A ∩ C = 70
B ∩ C = 50
A ∩ B ∩ C = ?
We can start by finding the total number of families that watch at least one sport:
Total = A + B + C - (A ∩ B) - (A ∩ C) - (B ∩ C) + (A ∩ B ∩ C)
Plugging in the values:
Total = 285 + 195 + 115 - 115 - 70 - 50 + (A ∩ B ∩ C)
Total = 360 + (A ∩ B ∩ C)
We are also given that 50 families do not watch any of the three sports.
Therefore, we have:
Total - 50 = 360 + (A ∩ B ∩ C) - 50
Simplifying the equation:
310 = 360 + (A ∩ B ∩ C)
Finally, to find the number of families that watch all three sports, we subtract 360 from both sides of the equation:
(A ∩ B ∩ C) = 310 - 360 = -50
Since negative values don't make sense in this context, it seems there was an error or inconsistency in the given information.