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: Sanchita wins if she gets a six on her turn. This requires Raj to fail all his turns before Sanchita succeeds. Model this as a geometric series: P(Sanchita wins) = 1/6 + (5/6)²·(1/6) + (5/6)⁴·(1/6) + ... = (1/6)/(1 - 25/36) = 6/11, then compute 5p = 5·(6/11) ≈ 2.27.
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