To calculate the probability of selecting one apple and one banana from the basket, we’ll first need to find the total number of fruits and then consider two scenarios: selecting an apple first or selecting a banana first.
Total fruits in the basket:
- Bananas: 5
- Mangoes: 8
- Apples: 7
Total fruits = 5 + 8 + 7 = 20 fruits
Scenario 1: Selecting an apple first, then a banana
- Probability of selecting an apple first: \[ P(\text{Apple first}) = \frac{7}{20} \]
- After selecting an apple, we have:
- 5 bananas
- 8 mangoes
- 6 apples (since we have taken 1 apple)
Total fruits left = 20 - 1 = 19
- Probability of selecting a banana next: \[ P(\text{Banana second | Apple first}) = \frac{5}{19} \]
So, the probability of selecting an apple first and then a banana is: \[ P(\text{Apple, then Banana}) = P(\text{Apple first}) \times P(\text{Banana second | Apple first}) = \frac{7}{20} \times \frac{5}{19} = \frac{35}{380} \]
Scenario 2: Selecting a banana first, then an apple
- Probability of selecting a banana first: \[ P(\text{Banana first}) = \frac{5}{20} = \frac{1}{4} \]
- After selecting a banana, we have:
- 4 bananas (since we have taken 1 banana)
- 8 mangoes
- 7 apples
Total fruits left = 20 - 1 = 19
- Probability of selecting an apple next: \[ P(\text{Apple second | Banana first}) = \frac{7}{19} \]
So, the probability of selecting a banana first and then an apple is: \[ P(\text{Banana, then Apple}) = P(\text{Banana first}) \times P(\text{Apple second | Banana first}) = \frac{1}{4} \times \frac{7}{19} = \frac{7}{76} \]
Total Probability
\[ P(\text{Apple and Banana}) = P(\text{Apple, then Banana}) + P(\text{Banana, then Apple}) = \frac{35}{380} + \frac{7}{76} \] First, convert \(\frac{7}{76}\) to have the same denominator as \(\frac{35}{380}\): \[ \frac{7}{76} = \frac{35}{380} \] Now, let's simplify: \[ \frac{35}{380} + \frac{35}{380} = \frac{70}{380} = \frac{7}{38} \]
Final Answer
The probability of selecting one apple and one banana is: \[ \frac{7}{38} \]
Since none of the provided options matches this probability, the closest option is \( \frac{7}{76} \) from the computation for selecting an apple followed by a banana.