To find the probability of selecting one apple and one banana from the basket without replacement, we can first determine the total number of fruits and then find the number of successful outcomes.
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Total Number of Fruits:
- Bananas: 5
- Mangoes: 8
- Apples: 7
- Total = 5 + 8 + 7 = 20 fruits
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Calculating the Probability: We can select one apple and one banana in two different orders: either an apple first and then a banana, or a banana first and then an apple.
Case 1: Apple first, then Banana
- Probability of choosing an apple first: \[ P(\text{apple first}) = \frac{7}{20} \]
- After choosing an apple, there are 19 fruits left. The probability of choosing a banana next: \[ P(\text{banana second | apple first}) = \frac{5}{19} \]
- So, the combined probability for this case is: \[ P(\text{Apple first, Banana second}) = P(\text{apple first}) \times P(\text{banana second | apple first}) = \frac{7}{20} \times \frac{5}{19} = \frac{35}{380} \]
Case 2: Banana first, then Apple
- Probability of choosing a banana first: \[ P(\text{banana first}) = \frac{5}{20} = \frac{1}{4} \]
- After choosing a banana, there are 19 fruits left. The probability of choosing an apple next: \[ P(\text{apple second | banana first}) = \frac{7}{19} \]
- So, the combined probability for this case is: \[ P(\text{Banana first, Apple second}) = P(\text{banana first}) \times P(\text{apple second | banana first}) = \frac{5}{20} \times \frac{7}{19} = \frac{35}{380} \]
- Total Probability: To find the total probability of selecting one apple and one banana (regardless of the order), we add both cases together: \[ P(\text{one apple and one banana}) = P(\text{Apple first, Banana second}) + P(\text{Banana first, Apple second}) = \frac{35}{380} + \frac{35}{380} = \frac{70}{380} \]
Now, simplify \(\frac{70}{380}\): \[ \frac{70}{380} = \frac{7}{38} \]
- Final Answer: The probability of selecting an apple and a banana is: \[ \frac{7}{38} \]
This probability does not appear in your listed responses. However, if you're looking for another way to express the result of selections or if there was a typo in your options, let me know!