Special Coin
Optiver
During a unique coin tossing competition, Sarah uses a special coin that lands heads with probability $p$, while Tom uses a standard fair coin. They take turns flipping their coins, with Sarah going first. The first person to get heads wins. If we know that both players have an equal chance of winning, what is the value of p?
Answer
Let $P_s$ be Sarah's probability of winning from her turn. From her position: She wins with probability $p$ (heads), or gets tails ($1-p$), then Tom gets tails ($\frac{1}{2}$) and we're back to Sarah's turn with probability $P_s$, or Tom gets heads and wins. This gives: $P_s = p + (1-p) \cdot \frac{1}{2} \cdot P_s$. For fair game, $P_s = \frac{1}{2}$. Solving: $\frac{1}{2} = p + (1-p) \cdot (\frac{1}{2})^2$.Therefore, $p = \frac{1}{3}$.