To find the probability that Jonas will get an orange and Beth will get an apple, we need to calculate the probability of each event happening in sequence.
Step 1: Calculate the probability that Jonas gets an orange.
The total number of pieces of fruit in the basket is 11 (4 apples + 5 oranges + 2 bananas).
The number of oranges is 5.
The probability that Jonas picks an orange is:
\[ P(\text{Jonas gets an orange}) = \frac{\text{Number of oranges}}{\text{Total number of fruits}} = \frac{5}{11} \]
Step 2: Calculate the probability that Beth gets an apple after Jonas has taken an orange.
After Jonas picks an orange, there are now 10 pieces of fruit left in the basket (4 apples, 4 oranges, and 2 bananas).
Now, the number of apples is still 4.
The probability that Beth picks an apple after Jonas has picked an orange is:
\[ P(\text{Beth gets an apple} \mid \text{Jonas got an orange}) = \frac{\text{Number of apples}}{\text{Total remaining fruits}} = \frac{4}{10} = \frac{2}{5} \]
Step 3: Calculate the overall probability that Jonas gets an orange and Beth gets an apple.
Since both events (Jonas picking an orange and Beth picking an apple) are independent events, we multiply their probabilities:
\[ P(\text{Jonas gets an orange and Beth gets an apple}) = P(\text{Jonas gets an orange}) \times P(\text{Beth gets an apple} \mid \text{Jonas got an orange}) \]
Substituting the probabilities we calculated:
\[ P(\text{Jonas gets an orange and Beth gets an apple}) = \frac{5}{11} \times \frac{2}{5} \]
Calculating this gives:
\[ = \frac{5 \times 2}{11 \times 5} = \frac{10}{55} = \frac{2}{11} \]
Thus, the final probability that Jonas will get an orange and Beth will get an apple is:
\[ \boxed{\frac{2}{11}} \]