A school is running a raffle for two prizes. 59 tickets were sold for the raffle, numbered 1,…,59. All the tickets are put into a hat and a teacher picks out two tickets which have numbers i and j from the hat. What is the expected value of |i−j|?
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To find the expected value of |i-j|, we need to consider all possible pairs of numbers i and j and calculate the probability of each pair being picked.
In this case, there are 59 tickets numbered from 1 to 59. The total number of pairs of tickets that can be picked is (59 choose 2), which is calculated as:
(59 choose 2) = (59!)/((2!)(57!)) = (59 x 58)/(2 x 1) = 1711
Now, let's calculate the probability for each possible value of |i-j|.
When |i-j| = 0:
There is only one pair where i and j are the same numbers. The probability of this happening is 1/1711.
When |i-j| = 1:
There are (58 choose 2) pairs where the difference between i and j is 1. This can occur if i is one less or one more than j. The probability of this happening is (58 x 57)/(2 x 1) = 1653/1711.
When |i-j| = 2:
There are (57 choose 2) pairs where the difference between i and j is 2. This can occur if i is two less or two more than j. The probability of this happening is (57 x 56)/(2 x 1) = 1596/1711.
Similarly, we can calculate the probabilities for |i-j| = 3, 4, 5, and so on.
Now, calculate the expected value by multiplying each possible value of |i-j| by its corresponding probability and summing them up:
Expected value = 0 x (1/1711) + 1 x (1653/1711) + 2 x (1596/1711) + ...
Continue this calculation for all possible values of |i-j|, up to the maximum difference of 58.
With the expected value formula, we arrive at the final answer.