If you have a six-sided die labeled 1-5 with two sides labeled 5, what's the probability of rolling the die five times and getting a unique side that you haven't previously rolled?
I've been mulling this one over for some time, and can't seem to get around the idea that the probability seems to change depending upon when you roll a 5. There must be a logical work around.
Best Answer
There are 5! = 120 distinct events to consider, one event corresponding to a different permutation of $\{1,\ldots,5\}$. Each such event has probability $$ \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \times \frac{1}{3}. $$ Why? Because $P(1) = P(2) = P(3)=P(4) = \frac{1}{6}$, $P(5)=\frac{1}{3}$, and on each of the 120 distinct events mentioned above, each of $\{1,2,3,4,5\}$ occurs exactly once independently of the other rolls. Thus, the probability of the event you describe is
$$ 120 \times \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} \times \frac{1}{3} = \frac{120}{6^4\cdot 3} = \frac{5}{162}. $$