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Chapter 1: Problem 83

Two distinct integers are chosen at random and without replacement from the first six positive integers. Compute the expected value of the absolute value of the difference of these two numbers.

Short Answer

Expert verified
The expected value of the absolute difference is \(2\).

Step by step solution

01

Identify possible absolute differences and their frequencies

Firstly, let's map out the six integers on a grid with each paired with every other. Since the chosen numbers are distinct, the pairs like (1,1),(2,2),(3,3),(4,4),(5,5), and (6,6) are excluded. We then calculate the absolute difference for each pair. The possible absolute differences are 1, 2, 3, 4 and 5. Count how many times each difference occurs. For instance, the difference '1' occurs 5 times - between (1,2), (2,3), (3,4), (4,5), and (5,6). Follow this method to get the frequencies of all the differences.
02

Calculate the probability of each absolute difference

Next, calculate the probability of each absolute difference by dividing the frequency of each difference by the total number of pairs. The total number of pairs is \( \binom{6}{2} = 15 \), or 6 choose 2, since we are selecting two numbers from six.
03

Compute expected value

With the probabilities, compute the expected value which is the sum of each absolute difference times its corresponding probability. E.g. `1 * P(1) + 2 * P(2) + 3 * P(3) + 4 * P(4) + 5 * P(5)`. Since the frequencies of the differences are already known from Step 1, and the probabilities from Step 2, use these to calculate the expected value.

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