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Median of a sample

ProbabilityMedium

$3$ distinct numbers are drawn without replacement, all subsets equally likely, from the integers $1$ to $10$ . What is the probability that the middle value (the median) equals $5$ ? Give a fraction in lowest terms.

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Hint

Count the subsets with one number below $5$ , the number $5$ itself, and one above.

Solution

There are $(3 10) = 120$ equally likely subsets.
For the median to be $5$ , the subset contains $5$ , one number from ${1, 2, 3, 4}$ and one from ${6, 7, 8, 9, 10}$ .
That gives $4 \times 5 = 20$ favourable subsets.
The probability is $\frac{20}{120} = \frac{1}{6}$ .
The answer is $\frac{1}{6}$ .

Answer: $\frac{1}{6}$

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