A library recorded whether each of 240 visitors participated in a reading challenge, attended an author event, both, or neither. Each visitor is listed in exactly one activity category.
| Visitor group | Neither activity | Reading challenge only | Author event only | Both activities | Total |
|---|---|---|---|---|---|
| Teen | 18 | 42 | ? | 30 | 120 |
| Adult | 36 | 15 | 45 | 24 | 120 |
| Total | 54 | 57 | 75 | 54 | 240 |
A visitor is selected at random from the visitors who participated in at least one of the two activities. What is the probability that the selected visitor is a teen and participated in exactly one of the two activities? (Express your answer as a fraction or decimal, not as a percent.)
For a conditional-probability question based on a table, first identify the group named after "from" or "given"; that group supplies the denominator. Next, count only people who meet both the condition and the event asked about for the numerator. When a table has a missing entry, complete it before counting so that no relevant category is overlooked.
Hints
Complete the Teen row
Use the Teen row total and its three known entries to find the missing entry.
Identify the conditional group
The phrase "participated in at least one" excludes only the visitors in the Neither activity column.
Identify exactly one activity
For teens, combine the two categories labeled "only" to count those who participated in exactly one activity.
Desmos Guide
Use arithmetic rather than graphing
Direct table arithmetic is faster for this question. Use Desmos to verify the calculations after identifying the relevant categories.
Check the missing entry
Enter 120-18-42-30 to find the missing number in the Teen row.
Verify the conditional probability
Enter (42+(120-18-42-30))/(240-54). The numerator represents teens in exactly one activity, and the denominator represents all visitors in at least one activity.
Step-by-step Explanation
Find the missing table entry
The Teen row has a total of visitors, so the number of teens who attended only the author event is
Count the visitors in the condition
Visitors who participated in at least one activity are everyone except those in the Neither activity column:
Count the favorable visitors
A teen who participated in exactly one activity is in either the Reading challenge only or Author event only category. The favorable count is
Therefore, the conditional probability is