Question 47·Medium·Probability and Conditional Probability
The counts of candies in a box are sorted by flavor and by the color of their wrappers, as shown in the table below.
| Flavor | Red wrapper | Blue wrapper | Total |
|---|---|---|---|
| Cherry | 24 | 16 | 40 |
| Lemon | 12 | 28 | 40 |
| Grape | 14 | 6 | 20 |
| Total | 50 | 50 | 100 |
If one candy is selected at random and is known to have a blue wrapper, what is the probability that it is lemon?
For two-way table probability questions, always start by identifying what is given in the question—that condition tells you which row or column becomes your new “total” (denominator). Then, within that restricted row or column, find the count that matches what you’re asked about (the numerator). Write the fraction as favorable over total within that condition, simplify it, and only then look at the answer choices to match the simplified form.
Hints
Focus on the right part of the table
Because the candy is known to have a blue wrapper, ignore everything except the Blue wrapper column.
Decide what counts as “total” now
Your total number of possible outcomes should be the total number of blue-wrapper candies, not the total number of candies in the box.
Find the favorable outcomes
In the blue-wrapper column, look at how many of those candies are lemon flavor. That number will be your numerator in the probability.
Form and simplify the fraction
Write the probability as (lemon & blue) over (total blue), then simplify the fraction to lowest terms before matching it to the answer choices.
Desmos Guide
Compute the conditional probability
In Desmos, type 28/50 to represent (lemon & blue) over (total blue-wrapper candies). Desmos will show the fraction and may simplify it automatically; note the simplified fraction it gives and match that to the answer choices.
Step-by-step Explanation
Identify what “given blue wrapper” means
The phrase “is known to have a blue wrapper” means we are only considering candies with blue wrappers.
So, instead of all 100 candies, our new “universe” is just the blue-wrapper column of the table.
Find the total number of blue-wrapper candies
Look at the Total row under Blue wrapper.
- Cherry, blue: 16
- Lemon, blue: 28
- Grape, blue: 6
- Total blue-wrapper candies:
So there are 50 candies that have blue wrappers.
Find how many of those blue candies are lemon
Now focus on the Lemon row under Blue wrapper.
- Lemon, blue: 28
These are the favorable outcomes: blue-wrapper candies that are lemon flavor.
Write and simplify the probability
The probability (given that the wrapper is blue) is:
- Favorable outcomes: 28 (lemon & blue)
- Possible outcomes (all blue): 50 (all blue-wrapper candies)
So the probability is
Now simplify this fraction by dividing numerator and denominator by 2:
So the correct answer is 14/25.