Question 5·Medium·Probability and Conditional Probability
The table below shows the number of state parks in a certain state that contain camping facilities and bicycle paths.
| Has bicycle paths | Does not have bicycle paths | |
|---|---|---|
| Has camping facilities | 20 | 5 |
| Does not have camping facilities | 8 | 4 |
If one of these state parks is selected at random, what is the probability that it has camping facilities but does not have bicycle paths?
For two-way table probability questions, first carefully identify the exact cell (combination of conditions) the question asks about and note that count as your numerator. Then, unless the question explicitly says "given that" or conditions on a row or column, use the sum of all entries in the table as your denominator. Finally, write probability as (favorable)/(total) and, if needed, simplify the fraction. Keeping numerator and denominator meanings clear helps avoid common conditional-probability traps.
Hints
Think about what counts as a favorable outcome
Focus only on parks that have camping facilities but do not have bicycle paths. Which single cell in the table shows that combination?
Find the total number of parks
To get the total number of parks, add all four numbers in the table. This will be the denominator of your probability.
Set up the probability fraction
Use the formula . Put the number from the specific cell you identified over the total you calculated.
Desmos Guide
Use Desmos to confirm the probability
In Desmos, type 5/(20+5+8+4) and press Enter. The output will be a fraction; this value is the probability from the table.
Step-by-step Explanation
Find the total number of parks
Add all the values in the table to find how many parks there are in total.
- Has camping and bike paths:
- Has camping, no bike paths:
- No camping, has bike paths:
- No camping, no bike paths:
Total number of parks:
Identify the favorable parks
We want the parks that have camping facilities but do not have bicycle paths.
From the table, that is the cell in the "Has camping facilities" row and the "Does not have bicycle paths" column, which is:
- Number of favorable parks:
Write the probability
Probability is
Here, the favorable outcomes are the 5 parks with camping and no bike paths, and the total outcomes are the 37 parks in the table. So the probability is
which matches answer choice D.