Question 6·Hard·Probability and Conditional Probability
Employees working for a customer service line at an electric company recorded all the calls last Monday and noted whether the caller asked for repairs and whether the caller asked about a bill. The results are summarized in the table below.
| Asked for repairs | Did not ask for repairs | Total | |
|---|---|---|---|
| Asked about a bill | 48 | 623 | 671 |
| Did not ask about a bill | 130 | 90 | 220 |
| Total | 178 | 713 | 891 |
If a caller last Monday who asked about his or her bill is selected at random, which of the following is closest to the probability that the customer also asked for repairs?
For conditional probability questions with tables, always first identify the given condition in the wording (for example, "who asked about his or her bill"). Restrict your attention to that row or column only—this gives you the denominator. Then find how many in that same row or column also satisfy the event of interest for the numerator. Form the fraction (numerator over denominator), compute or approximate the decimal (using a calculator if allowed), and choose the closest answer. Be careful not to default to using the grand total from the table unless the question clearly asks about all cases.
Hints
Focus on the right group
Pay close attention to the phrase "a caller ... who asked about his or her bill." Which row of the table represents callers who asked about a bill?
Decide what goes in the denominator
For conditional probability, the denominator is only the total number of outcomes that satisfy the given condition. What is the total number of callers who asked about a bill?
Decide what goes in the numerator
Among the callers who asked about a bill, how many also asked for repairs? Look at the correct cell at the intersection of the appropriate row and column.
Turn the fraction into a decimal
Once you have the fraction (a smaller number over a larger number), use division or a calculator to get a decimal and then see which answer choice it is closest to.
Desmos Guide
Compute the conditional probability
In Desmos, type 48/671 and look at the decimal value it gives. Then match that decimal to the closest answer choice in the problem.
Step-by-step Explanation
Identify the condition (the restricted group)
The question says: "If a caller last Monday who asked about his or her bill is selected at random".
This means we are only looking at callers who asked about a bill. In the table, that is the first row, which has a total of 671 callers.
Find how many in this group also asked for repairs
Within the row "Asked about a bill," look at the entry under the column "Asked for repairs."
That cell is 48, meaning 48 callers both:
- asked about a bill, and
- asked for repairs.
Write the conditional probability as a fraction
We want the probability that a caller also asked for repairs, given that they asked about a bill.
So the probability is:
Convert the fraction to a decimal and match the choice
Now approximate .
Using a calculator:
Rounded to two decimal places, this is about , which corresponds to choice B) 0.07.