To find the probability that a randomly selected box of cereal is jumbo size given that the box contains a prize, we use the concept of conditional probability.
Let:
- A = the event that a box is jumbo size.
- B = the event that a box contains a prize.
We want to find \( P(A | B) \), the probability that a box is jumbo size given that it contains a prize. This can be calculated using the formula:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
Step 1: Find \( P(A \cap B) \) (the probability that a box is both jumbo size and contains a prize):
From the given data, the number of jumbo boxes that contain a prize is 7.
Step 2: Find \( P(B) \) (the total probability that a box contains a prize):
To find \( P(B) \), we need to sum the boxes that contain a prize across all sizes:
- Mini with prize: 9
- Regular with prize: 5
- Jumbo with prize: 7
Total boxes with a prize = \( 9 + 5 + 7 = 21 \).
Step 3: Find the total number of boxes:
Now, let's calculate the total number of boxes, both with and without a prize:
- Mini without prize: 6
- Regular without prize: 9
- Jumbo without prize: 8
Total boxes = \( 9 + 6 + 5 + 9 + 7 + 8 = 44 \).
Step 4: Calculate \( P(B) \):
Now we have the total boxes with a prize (21) and the total number of boxes (44):
\[ P(B) = \frac{\text{Total with prize}}{\text{Total boxes}} = \frac{21}{44} \]
Step 5: Calculate \( P(A \cap B) \):
The probability that a box is jumbo and contains a prize is the favorable outcome (7) divided by total boxes (44):
\[ P(A \cap B) = \frac{7}{44} \]
Step 6: Calculate \( P(A | B) \):
Now we can substitute back into our conditional probability formula:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{7}{44}}{\frac{21}{44}} = \frac{7}{21} = \frac{1}{3} \]
Thus, the probability that a randomly selected box of cereal is jumbo size given that the box contains a prize is \( \frac{1}{3} \).