Math or magic ?
Let there be some apples and some oranges in a basket. Let a man picks two fruits one after another at random. In one case he picks the fruits with replacement, and in another he picks the fruits without replacement. Let the probabilities that the second fruit picked is an orange in the two cases be and respectively. Find
The answer is 15.
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1 solution
The probability that the second fruit is an orange, unconditioned on any knowledge of the type of the first fruit, is the same in both cases. This is true no matter how many of each fruit there are.
Indeed, the (unconditioned) probability that the second fruit is an orange is identical to the (unconditioned) probability that the first fruit is an orange.
If that's not convincing....
Let's say there are m apples and n oranges. Clearly, if replacement is allowed, the probability of the second fruit being an orange is m + n n .
If replacement isn't allowed, then there are two cases. In the first case, the first fruit is an apple and the second is an orange. This event happens with probabiliy m + n m × m + n − 1 n . The second case is when the both fruit are oranges. This event happens with probability m + n n × m + n − 1 n − 1 . So the total probability is the sum of these two partial probabilities:
m + n m × m + n − 1 n + m + n n × m + n − 1 n − 1 = m + n n × ( m + n − 1 m + m + n − 1 n − 1 ) = m + n n
Thus the two probabilities are equal.