Probability of picking strawberry ice cream for first time on the sixth selection from five flavors?

Question

An ice cream store has five flavors. If we pick flavors successively selected at random, what is the probability that the flavor strawberry will be selected for the first time on the sixth selection?

Answer 1

If the strawberry flavor was chosen for the first time on the sixth iteration this means, we did not choose it on previous $5$ iterations. The probability of not to choose the strawberry flavor at random pick is $$ \frac{4}{5}, $$ i.e. $4$ possibilities out of $5$ are not strawberry. We did this $5$ times in succession, thus the probability of not to choose the strawberry flavor at any of $5$ trials is $$ \left(\frac{4}{5}\right)^5. $$

At the sixth time we succeeded to choose the strawberry flavor. The probability of this event was $$ \frac{1}{5}, $$ i.e. one possibility out of $5$.

Now collect everything together. The probability of 5 "misses" and then a "hit" is the multiple $$ \left(\frac{4}{5}\right)^5\frac{1}{5} = \frac{4^5}{5^6} \approx 6.5\% $$