To calculate the probability of selecting an apple and a banana from the basket of fruits, we can follow these steps:
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Identify the total number of fruits:
- Total fruits = 5 bananas + 8 mangoes + 7 apples = 20 fruits
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Calculate the number of favorable outcomes for selecting 1 apple and 1 banana:
- We can have two scenarios for selecting an apple and a banana: selecting a banana first and then an apple, or selecting an apple first and then a banana.
Scenario 1: Select a banana first, then an apple
- Probability of selecting a banana first: \( \frac{5}{20} \)
- After selecting a banana, the number of remaining fruits: 19 (4 bananas, 8 mangoes, 7 apples)
- Probability of selecting an apple next: \( \frac{7}{19} \)
- Combined probability for this scenario: \[ P(\text{banana first, then apple}) = \frac{5}{20} \times \frac{7}{19} = \frac{5 \times 7}{20 \times 19} = \frac{35}{380} \]
Scenario 2: Select an apple first, then a banana
- Probability of selecting an apple first: \( \frac{7}{20} \)
- After selecting an apple, the number of remaining fruits: 19 (5 bananas, 8 mangoes, 6 apples)
- Probability of selecting a banana next: \( \frac{5}{19} \)
- Combined probability for this scenario: \[ P(\text{apple first, then banana}) = \frac{7}{20} \times \frac{5}{19} = \frac{7 \times 5}{20 \times 19} = \frac{35}{380} \]
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Total probability of selecting one apple and one banana:
- Add the probabilities from both scenarios: \[ P(\text{one apple and one banana}) = P(\text{banana first, then apple}) + P(\text{apple first, then banana}) = \frac{35}{380} + \frac{35}{380} = \frac{70}{380} \]
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Simplify the probability:
- \(\frac{70}{380} = \frac{7}{38}\)
Therefore, the probability of selecting one apple and one banana is \( \frac{7}{38} \).