To find the probability that both fruits Sharon took are apples, we can use the concept of combinations.
The total number of fruits in the basket is:
- 5 apples + 3 pears = 8 fruits.
Now, we want to find the probability that both fruits taken are apples.
- Calculate the total number of ways to choose 2 fruits from the 8 fruits. This can be done using combinations, denoted as \( C(n, k) \), which represents the number of ways to choose \( k \) items from \( n \) items without regard to the order of selection.
\[ C(8, 2) = \frac{8!}{2!(8-2)!} = \frac{8 \times 7}{2 \times 1} = 28. \]
- Next, we calculate the number of ways to choose 2 apples from the 5 apples:
\[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10. \]
- Now, we can determine the probability that both fruits taken by Sharon are apples. The probability \( P \) is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(\text{both fruits are apples}) = \frac{C(5, 2)}{C(8, 2)} = \frac{10}{28}. \]
- We can simplify this fraction:
\[ \frac{10}{28} = \frac{5}{14}. \]
Thus, the probability that both fruits Sharon took are apples is:
\[ \frac{5}{14}. \]