Question 15·Hard·Probability and Conditional Probability
In a park, a visitor is assigned a table by randomly selecting one card from a set of 54 equally likely assignment cards. Each card shows the picnic area (A, B, or C) and whether the assigned table has an umbrella, as shown below.
| Picnic area | Umbrella | No umbrella |
|---|---|---|
| A | 6 | 12 |
| B | 9 | 9 |
| C | 14 | 4 |
Given that the assigned table has an umbrella, what is the probability that it is in Area C?
For conditional probability from a two-way table, restrict to the row/column named in the “given” statement, then compute
Here, “given umbrella” means the umbrella column becomes the new total, and you take the Area C umbrella count over that umbrella total.
Hints
Focus on the condition
Because the table is known to have an umbrella, ignore the “No umbrella” column.
Find the new total
Add the umbrella counts across A, B, and C to get the total number of umbrella outcomes.
Make a ratio
Your probability is
Desmos Guide
Compute the conditional probability from the umbrella column
In Desmos, enter:
14/(6+9+14)
This is the fraction of umbrella outcomes that are in Area C.
Match to the answer choices
Convert/simplify the Desmos result to a fraction and match it to the listed options.
Step-by-step Explanation
Use the given condition (umbrella) to narrow the sample space
Since we are given that the assigned table has an umbrella, we only look at the Umbrella column of the table.
Count total umbrella outcomes and umbrella outcomes in Area C
Total umbrella outcomes:
Umbrella outcomes in Area C: .
Form the conditional probability as a proportion
Given umbrella, the probability the area is C is
So the correct answer is .