Final answer:
The probability of all four children receiving the same fruit can be found by adding the probability of all receiving apples with the probability of all receiving oranges, which are calculated by multiplying the probabilities of each child receiving the same fruit in succession.
Step-by-step explanation:
The question asks about the probability that all four children receive the same type of fruit (either all apples or all oranges) from a box when picked randomly without replacement. To find the probability of such an event, we must consider the two scenarios separately: receiving all apples or receiving all oranges.
For all children to receive an apple, the first child has a 5/11 chance (since there are 5 apples and 6 oranges, making a total of 11 fruits). Without replacement, the second child would then have a 4/10 chance, the third child a 3/9, and the fourth child a 2/8 chance. Multiplying these probabilities together gives us the probability for all children receiving an apple, (5/11) * (4/10) * (3/9) * (2/8).
For all children to receive an orange, we perform a similar calculation, starting with a 6/11 chance for the first child, then 5/10, 4/9, and 3/8 for subsequent children. The probability is then (6/11) * (5/10) * (4/9) * (3/8).
To find the total probability of all children receiving the same type of fruit, we add the two probabilities together.