Question

In a survey, 672 adults were asked what television programs they had recently watched. The following information was obtained: 226 adults watched neither the Big Game nor the New Movie, and 287 watched the New Movie. If 183 of those who watched the New Movie did not watch the Big Game, how many of the surveyed adults watched the following?

a. the Big Game.
b. the Big Game but not the New Movie.
c. at least one program.
d. both programs.

Answer 1

a. the Big Game = 263

b. the Big Game but not the New Movie = 159

c. at least one program = 446

d. both programs = 104.

Explanation:

Given that:

Total adults asked about TV programs, n(U) = 672

Total adults who watch New Movie, n(A) = 287

Let n(B) be the number of adults who watch Big Game.

Total adults who watch New Movie but not Big Game, n(A-B) = 183

As per formula:

n(A-B) = n(A) - n(A
`\cap` B)

Where n(A
`\cap` B) is the number of adults who watch New Movie and Big Game both.

n(A
`\cap` B) = n(A) - n(A-B)

n(A
`\cap` B) = 287 - 183 = 104

Formula:

n(U) = n(A
`\cup` B) + n(A
`\cup` B)'

n(A
`\cup` B) is the number of adults who watch at least one program.

n(A
`\cup` B)' is the number of adults who do not watch any program.

Given that n(A
`\cup` B)' = 226

Putting values in the above formula:

672 = n(A
`\cup` B) + 226

n(A
`\cup` B) = 672 - 226 = 446

Formula:

n(A
`\cup` B) = n(A) + n(B) - n(A
`\cap` B)

446 = 287 + n(B) - 104

n(B) = 446 + 104 - 287

n(B) = 263

The number of adults who watch the Big Game but not the New Movie, n(B - A)

Formula:

n(B - A) = n(B) - n(A
`\cap` B)

n(B - A) = 263 - 104 = 159

So, the number of surveyed adults who watch:

a. the Big Game, n(A) = 263

b. the Big Game but not the New Movie, n(B - A) = 159

c. at least one program, n(A
`\cup` B) = 446

d. both programs, n(A
`\cap` B) = 104.