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Smallest of $3$ draws

ProbabilityEasy

$3$ numbers are drawn independently, each uniformly from the integers $1$ to $10$ . What is the probability that the smallest of them is at least $8$ ? Give a fraction in lowest terms.

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Hint

The minimum is at least a threshold only if all draws are.

Solution

The smallest is at least $8$ exactly when all $3$ draws lie in ${8, 9, 10}$ .
One draw lies there with probability $\frac{3}{10}$ .
By independence the probability is $(\frac{3}{10})^{3} = \frac{27}{1000}$ .
The answer is $\frac{27}{1000}$ .

Answer: $\frac{27}{1000}$

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