Probability
Coin and die — probability die shows 1 after 5 steps
MJAT_TS4_P1
Grade 12
Question:
A fair coin and a fair 6-sided die are used. The die starts with 1 face up. A step: toss the coin — if heads, roll the die; if tails, do nothing. After 5 such steps, the probability that 1 is face up is $\alpha/\beta$ (in lowest terms). Then $\alpha+\beta=$
Step-by-Step Solution
Key Concept: P(die shows 1 | $k$ rolls have occurred) $= 1/6$ for any $k\geq 1$, and $= 1$ for $k=0$. P($k$ rolls in 5 steps) $= \binom{5}{k}(1/2)^k(1/2)^{5-k} = \binom{5}{k}/32$. So $P(\text{die shows 1}) = \frac{1}{32}\sum_{k=0}^5\binom{5}{k}\cdot P(\text{shows 1}|k\text{ rolls})$.
$P=\frac{37}{192}$. $\alpha+\beta=37+192=\mathbf{229}$.
Correct Answer: 229