To calculate the probability of selecting one apple and one banana from a basket containing 5 bananas, 8 mangoes, and 7 apples, we first determine the total number of fruits in the basket.
Total fruits = 5 (bananas) + 8 (mangoes) + 7 (apples) = 20 fruits.
Now, we want to find the probability of selecting one apple and one banana in two draws without replacement.
There are two possible orders for selecting one apple and one banana:
- Select an apple first, then a banana.
- Select a banana first, then an apple.
Case 1: Apple first, then banana
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The probability of selecting an apple first: \[ P(\text{Apple first}) = \frac{7}{20} \]
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After selecting an apple, there are now 19 fruits left, including 5 bananas. The probability of then selecting a banana: \[ P(\text{Banana second | Apple first}) = \frac{5}{19} \]
The combined probability for this case: \[ P(\text{Apple first, Banana second}) = P(\text{Apple first}) \times P(\text{Banana second | Apple first}) = \frac{7}{20} \times \frac{5}{19} = \frac{35}{380} \]
Case 2: Banana first, then apple
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The probability of selecting a banana first: \[ P(\text{Banana first}) = \frac{5}{20} = \frac{1}{4} \]
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After selecting a banana, there are 19 fruits left, including 7 apples. The probability of selecting an apple: \[ P(\text{Apple second | Banana first}) = \frac{7}{19} \]
The combined probability for this case: \[ P(\text{Banana first, Apple second}) = P(\text{Banana first}) \times P(\text{Apple second | Banana first}) = \frac{5}{20} \times \frac{7}{19} = \frac{35}{380} \]
Total Probability
Now, we add the probabilities from both cases: \[ P(\text{One Apple and One Banana}) = P(\text{Apple first, Banana second}) + P(\text{Banana first, Apple second}) = \frac{35}{380} + \frac{35}{380} = \frac{70}{380} \]
This simplifies to: \[ \frac{70}{380} = \frac{7}{38} \]
None of the responses match this result, but among the options you indicated, it seems you may have other simplifying to do.
None of the provided answers seem to match \(\frac{7}{38}\). However, if we go through simplifications or changes in context, we might find some ways to recalculate steps through approximation or simple fraction addition to relate it back to your choices more closely. Check values or input to validate your final answer.