(14) In a school, [tex]$\frac{3}{5}$[/tex] of the staff read Daily Times and [tex]$\frac{1}{2}$[/tex] read Tribune and [tex]$\frac{1}{10}$[/tex] read neither. What fraction of the staff read both types of papers?
Answer (1)
Calculate the probability of reading at least one newspaper: P ( A c u pB ) = 1 − 10 1 = 10 9 .
Use the inclusion-exclusion principle: P ( A c a pB ) = P ( A ) + P ( B ) − P ( A c u pB ) .
Substitute the given values: P ( A c a pB ) = 5 3 + 2 1 − 10 9 .
Calculate the fraction of staff reading both: P ( A c a pB ) = 5 1 .
Explanation
Analyze the problem and data We are given the fractions of staff who read the Daily Times, the Tribune, and neither paper. We need to find the fraction of staff who read both papers. Let's denote the fraction of staff who read Daily Times as P ( A ) , the fraction who read Tribune as P ( B ) , and the fraction who read neither as P ( A ′ c a p B ′ ) .
List given probabilities We have the following information:
P ( A ) = 5 3 (Daily Times)
P ( B ) = 2 1 (Tribune)
P ( A ′ c a p B ′ ) = 10 1 (Neither)
Calculate probability of reading at least one paper We know that the fraction of staff who read neither paper is the complement of the fraction who read at least one paper (Daily Times or Tribune). Therefore,
P ( A c u pB ) = 1 − P ( A ′ c a p B ′ ) = 1 − 10 1 = 10 9
Apply the inclusion-exclusion principle We also know that the probability of A c u pB can be expressed as:
P ( A c u pB ) = P ( A ) + P ( B ) − P ( A c a pB )
Rearrange the formula to find the intersection We want to find P ( A c a pB ) , which represents the fraction of staff who read both papers. Rearranging the formula, we get:
P ( A c a pB ) = P ( A ) + P ( B ) − P ( A c u pB )
Substitute the values Now, substitute the given values:
P ( A c a pB ) = 5 3 + 2 1 − 10 9
Calculate the final probability To solve this, we need a common denominator, which is 10:
P ( A c a pB ) = 10 6 + 10 5 − 10 9 = 10 6 + 5 − 9 = 10 2 = 5 1
State the final answer Therefore, the fraction of the staff who read both types of papers is 5 1 .
Examples
Understanding the readership overlap between different newspapers can help a school's administration decide where to place advertisements to reach the most staff members. For example, if a school wants to announce a staff meeting, knowing that 5 1 of the staff reads both Daily Times and Tribune allows them to estimate the reach of announcements placed in either or both newspapers. This helps in optimizing communication strategies and resource allocation.
