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In the resample or not

ProbabilityMedium

A bootstrap resample draws $4$ times with replacement from $4$ different observations. What is the probability that one particular observation appears at least once? Answer as a fraction in lowest terms.

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Hint

Find the chance that it never appears, then take the complement.

Solution

One draw misses it with probability $\frac{3}{4}$ , so all draws miss it with probability $(\frac{3}{4})^{4} = \frac{81}{256}$ .
The complement is $1 - \frac{81}{256} = \frac{175}{256}$ , so the answer is $\frac{175}{256}$ .

Answer: $\frac{175}{256}$

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