To find the probability that the first person chooses a round fruit (either an apple or an orange), and the second person chooses a banana, we can follow these steps:
Step 1: Count the Fruits
- Apples: 3
- Oranges: 5
- Bananas: 4
- Total fruits: \(3 + 5 + 4 = 12\)
Step 2: Identify Round Fruits
The round fruits in the bag are apples and oranges:
- Total round fruits: \(3 + 5 = 8\)
Step 3: Calculate the Probability of the First Choice
The probability that the first person chooses a round fruit is: \[ P(\text{First chooses round fruit}) = \frac{\text{Number of round fruits}}{\text{Total fruits}} = \frac{8}{12} = \frac{2}{3} \]
Step 4: Calculate the Probability of the Second Choice
After the first person has chosen a fruit, we need to determine the remaining fruits to calculate the probability for the second choice. There are two cases: the first person chooses a round fruit or a banana.
- If the first person chooses a round fruit (8/12):
- Remaining fruits: 11 (12 - 1)
- Remaining bananas: 4
- Probability the second person chooses a banana: \[ P(\text{Second chooses banana | First chose round}) = \frac{4}{11} \]
Step 5: Combined Probability for First Case
To find the combined probability for this scenario (first person chooses a round fruit and the second chooses a banana): \[ P(\text{First round, Second banana}) = P(\text{First round}) \times P(\text{Second banana | First round}) = \frac{2}{3} \times \frac{4}{11} = \frac{8}{33} \]
Thus, the probability that the first person chooses a round fruit and the second chooses a banana is: \[ \boxed{\frac{8}{33}} \]