Question 23·Medium·Probability and Conditional Probability
Solar Panel Installation on 25 Homes
| Type of home | Solar panels installed | No solar panels | Total |
|---|---|---|---|
| Detached | 6 | 9 | 15 |
| Townhouse | 3 | 7 | 10 |
| Total | 9 | 16 | 25 |
A real-estate company surveyed 25 recently sold homes and recorded whether each home was a detached house or a townhouse and whether it had solar panels installed. The results are summarized in the table above. If one of the 25 homes is selected at random, what is the probability that the home selected will be a townhouse with solar panels installed?
For probability questions using a two-way table, first read the question phrase carefully to identify the exact row-and-column cell you need (for example, “townhouse with solar panels”), then take that cell’s count as the numerator and the overall total at the bottom-right of the table as the denominator, unless the question explicitly says something like “given that it is a townhouse,” which would change the denominator to a row or column total. This quick scan—cell, then total—helps you avoid common traps that use row totals or column totals incorrectly.
Hints
Locate the right cell in the table
You are looking for homes that are both townhouses and have solar panels installed. Which single cell in the table shows that combination?
Count favorable outcomes and total outcomes
How many homes are townhouses with solar panels (from that one cell)? How many homes are there in total (from the bottom-right of the table)?
Turn counts into a probability
Probability is a fraction: number of favorable homes over total homes. Put the townhouse-with-solar count in the numerator and 25 in the denominator, then see which answer choice matches that fraction.
Desmos Guide
Check the probability as a decimal or fraction
In a new expression line, type a fraction whose numerator is the number of townhouses with solar panels (from the table) and whose denominator is 25, like numerator/25. Desmos will show a decimal and a simplified fraction; compare that fraction to the answer choices to see which one matches.
Step-by-step Explanation
Identify the total number of possible homes
The probability is based on all homes that could be chosen. From the table, the total number of homes surveyed is in the bottom-right corner: .
So there are possible outcomes when one home is selected at random.
Find the number of favorable homes
We want the probability that the selected home is a townhouse with solar panels installed.
Look at the row for Townhouse and the column for Solar panels installed. The number in that cell is the number of favorable homes.
From the table, that cell shows townhouses with solar panels installed.
Write the probability as a fraction
Probability is
Here, the number of favorable outcomes is (townhouses with solar panels) and the total number of homes is .
So the probability is
which matches answer choice A.