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ProbabilityHard

$3$ distinct numbers are drawn without replacement, all subsets equally likely, from the integers $1$ to $8$ . What is the expected value of the largest number drawn? Give a fraction in lowest terms.

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Hint

The largest equals $m$ when the other $2$ numbers come from ${1, ..., m - 1}$ .

Solution

There are $(3 8) = 56$ equally likely subsets.
The largest equals $m$ in $(2 m - 1)$ subsets: $1, 3, 6, 10, 15, 21$ for $m = 3, ..., 8$ .
The sum of the products $m \times (2 m - 1)$ is $3 + 12 + 30 + 60 + 105 + 168 = 378$ .
The expected value is $\frac{378}{56}$ .
In lowest terms the answer is $\frac{27}{4}$ .

Answer: $\frac{27}{4}$

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