Fruity
The Brilliant Fruit Company has a surplus of apples and oranges. They decide to try some novel marketing by selling Mystery Fruit Bags. Each bag contains pieces of fruit, and any of the possible mixtures of apples and oranges. If Jane buys a Mystery Fruit Bag, pulls out a piece of fruit and finds it is an apple, what is the probability that the next piece of fruit she pulls out at random is also an apple?
If the answer can be expressed in the form where and are co-prime positive integers, give your answer as .
The answer is 5.
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1 solution
The probability of any particular mixture is N + 1 1 . Among the N + 1 possible mixtures there is a total of 2 N ( N + 1 ) apples, each of which has an equal probability N ( N + 1 ) 2 of being picked first, given that the first piece of fruit picked is an apple. The probability that the first apple came from a bag with k apples is N ( N + 1 ) 2 k . After the first apple is removed, the probability of picking a second apple from this bag is N − 1 k − 1 . The probability that the first apple came from that bag, and the second piece of fruit is also an apple is just the product of these two probabilities, and so is
N ( N + 1 ) ( N − 1 ) 2 k ( k − 1 ) .
So the overall probability is
k = 0 ∑ N N ( N + 1 ) ( N − 1 ) 2 k ( k − 1 ) = N ( N + 1 ) ( N − 1 ) 2 k = 0 ∑ N ( k 2 − k ) = N − 1 2 ( 6 2 N + 1 − 2 1 ) = 3 2 .
Hence the answer is 2 + 3 = 5 .