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Full range of $3$ spins

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A fair spinner shows the numbers $1$ , $2$ , $3$ , $4$ with equal probability. It is spun $3$ times independently. What is the probability that the smallest result is $1$ and the largest is $4$ ? Give a fraction in lowest terms.

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Hint

Use inclusion-exclusion on the outcomes missing $1$ or missing $4$ .

Solution

There are $4^{3} = 64$ equally likely outcomes.
Outcomes with no $1$ number $3^{3} = 27$ , and likewise $27$ have no $4$ .
Outcomes with neither $1$ nor $4$ number $2^{3} = 8$ .
Favourable outcomes: $64 - 27 - 27 + 8 = 18$ .
The probability is $\frac{18}{64} = \frac{9}{32}$ .

Answer: $\frac{9}{32}$

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