We are given the survey data of 750 people reading different magazines, and we need to answer three questions related to probability. Let's break this down step by step, starting by defining some sets for clarity:

• Let $T$ represent the people who read Time.
• Let $N$ represent the people who read Newsweek.
• Let $U$ represent the people who read U.S. News & World Report.

Given Data:

• Total people: 750
• $|T| = 230$ (people who read Time)
• $|N| = 180$ (people who read Newsweek)
• $|U| =$ (unknown, we'll calculate this)
• $|T \cap N| = 100$ (people who read both Time and Newsweek)
• $|T \cap U| = 90$ (people who read both Time and U.S. News & World Report)
• $|N \cap U| = 80$ (people who read both Newsweek and U.S. News & World Report)
• $|T \cap N \cap U| = 50$ (people who read all three magazines)
• $420$ people read none of these magazines.

(a) Find the probability that a randomly selected person reads U.S. News & World Report.

Using the inclusion-exclusion principle, we can calculate the number of people who read U.S. News & World Report.

Step 1: Inclusion-Exclusion Formula

|U| = (|T \cap U| + |N \cap U|) - |T \cap N \cap U| + \text{(People who read only U.S. News & World Report)} But first, we need to calculate how many people read exactly one magazine (this will help us find those who only read U.S. News & World Report).

Now let's compute each part carefully:

• People who read only Time: $|T| - |T \cap N| - |T \cap U| + |T \cap N \cap U| = 230 - 100 - 90 + 50 = 90$
• People who read only Newsweek: $|N| - |T \cap N| - |N \cap U| + |T \cap N \cap U| = 180 - 100 - 80 + 50 = 50$
• People who read only U.S. News & World Report: $\text{Total people} - (\text{people who read only Time, Newsweek, both, or all three}) - 420 =$