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Range of $3$ tickets

Expected valueMedium

$3$ different tickets are drawn at random, without replacement, from tickets numbered $1$ to $10$ . What is the expected value of the largest number minus the smallest number?

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Hint

For a random subset, the expected extremes have simple formulas.

Solution

For $k$ distinct numbers from $1$ to $n$ , the expected maximum is $\frac{k ( n + 1 )}{k + 1}$ and the expected minimum is $\frac{n + 1}{k + 1}$ .
With $k = 3$ and $n = 10$ these are $\frac{33}{4}$ and $\frac{11}{4}$ .
The expected range is $\frac{33}{4} - \frac{11}{4} = \frac{11}{2}$ .

Answer: $\frac{11}{2}$

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