Coin Toss and Dice Roll
A fair coin and a fair 6-sided die are used. The die begins showing 1. Each step starts with a coin toss: if the coin is heads, we roll the die; if tails, we do nothing else in that step. After 5 steps, what is the probability that the die still shows 1?
Answer
Two cases: Case 1: All 5 coin tosses come up tails (probability \(\tfrac{1}{32}\)). Then the die is never rolled and remains on 1, so this contributes
\[
\tfrac{1}{32}\times 1 = \tfrac{1}{32}.
\]
Case 2: At least one coin toss is heads (probability \(\tfrac{31}{32}\)). By the end, the die will have been rolled at least once. Each roll is fair, so the face is equally likely to be any number from 1 to 6. Thus the probability of ending on 1 is \(\tfrac{1}{6}\), contributing
\[
\tfrac{31}{32}\times \tfrac{1}{6} = \tfrac{31}{192}.
\]
Adding these two cases gives the total probability:
\[
\tfrac{1}{32} \;+\; \tfrac{31}{192}
= \tfrac{6}{192} \;+\; \tfrac{31}{192}
= \tfrac{37}{192}.
\]