Raj and Sanchita are playing game in which they throw a die alternately till one of them gets a six. The probability that Sanchita win the game is $p$ then the value of $5p?$
Step-by-Step Solution
Key Concept: A geometric series models alternating turns where one player wins on their turn with probability $p$ and both pass with probability $(1-p)^2$.
Sanchita wins if she rolls a six first. The probability of getting a six is $1/6$. If Sanchita starts, the probability she wins is $\frac{1}{6} + \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{1}{6} + \ldots = \frac{1/6}{1-25/36} = \frac{6}{11}$. If Raj starts, the probability that Sanchita wins is $1 - \frac{6}{11} = \frac{5}{11}$.
Correct Answer: 2.27