To solve this problem, we need to determine the probability that both mangoes picked are good. We'll do this step by step:
1. Determine the total number of mangoes and the number of good mangoes:
- Total mangoes = 8
- Bad mangoes = 3
- Good mangoes = [tex]\(8 - 3 = 5\)[/tex]
2. Calculate the probability that the first mango picked is good:
- Probability of picking a good mango first = [tex]\(\frac{\text{Number of good mangoes}}{\text{Total number of mangoes}} = \frac{5}{8}\)[/tex]
3. Calculate the probability that the second mango picked is good, given that the first mango picked was good:
- If the first mango picked is good, then there are now 4 good mangoes left out of the remaining 7 mangoes.
- Probability of picking a good mango second, given the first was good = [tex]\(\frac{\text{Remaining good mangoes}}{\text{Remaining total mangoes}} = \frac{4}{7}\)[/tex]
4. Calculate the overall probability that both picks are good:
- The probability that both mangoes picked are good is the product of the probabilities of each event (the first being good and the second being good given the first was good):
- Probability both are good = [tex]\(\frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}\)[/tex]
So, the probability that both mangoes picked are good is [tex]\(\frac{5}{14}\)[/tex], which corresponds to Option 1.
- Incorrect option:
- [tex]\(\frac{2}{7}\)[/tex]
Hence, the correct answer is [tex]\(\boxed{\frac{5}{14}}\)[/tex].
