Question 45·Hard·Probability and Conditional Probability
A biology lab stores petri dishes with bacterial cultures in three refrigerators, labeled A, B, and C. The table shows the number of dishes of each strain in each refrigerator.
| Refrigerator | Strain X | Strain Y | Total |
|---|---|---|---|
| A | 7 | 5 | 12 |
| B | 6 | 12 | 18 |
| C | 4 | 6 | 10 |
| Total | 17 | 23 | 40 |
One dish is selected at random from all 40 dishes. If the selected dish contains strain X, what is the probability that it came from refrigerator A?
For conditional probability from a two-way table, always restrict the “total” to the group named after the word given (here, strain X). Then compute the ratio (count in the desired category within that group) ÷ (total count in that group).
Hints
Focus on the condition "strain X"
Since the dish is known to be strain X, pretend the only dishes that exist are the strain X dishes.
Find the total number of strain X dishes
Add the strain X counts from refrigerators A, B, and C to get the total number of strain X dishes.
Form the ratio for a conditional probability
Compute
Desmos Guide
Compute the total number of strain X dishes
In Desmos, enter totalX = 7 + 6 + 4.
Compute the conditional probability
Enter 7/totalX and convert to a fraction if needed.
Match to the answer choice
Select the answer choice that matches the fraction shown by Desmos.
Step-by-step Explanation
Restrict to the strain X dishes
The condition “the selected dish contains strain X” means the relevant total is the number of strain X dishes, not the total number of dishes.
From the table, the total number of strain X dishes is
Use the conditional probability as a within-group ratio
Given that the dish is strain X, the probability it came from A is
State the answer
Therefore, the probability is .