Six-First Challenge

Solution & Explanation

Problem Statement

In a game where Amy and Brad alternate rolling a standard six-sided die, the first player to roll a '6' wins. Amy rolls first. We need to determine the probability that Amy wins.

Approach

The solution involves understanding the sequence of events and probabilities associated with each player's turn. Each player has a 1/6 chance of rolling a '6' on any given roll.

Linear Equations Approach

1. Definitions:

   • Let $P(A)$ be the probability that Amy wins.
   • Let $P(B)$ be the probability that Brad wins.
   • $P(F)$ is the probability that the first player wins, i.e., $P(F) = P(A)$ at the start.

2. Equations:

   • If Amy loses the first roll (does not roll a '6'), Brad becomes the "first" player. Hence, $P(B) = \left(\frac{5}{6}\right) \cdot P(A)$.
   • Since one of them must win, $P(A) + P(B) = 1$.

3. Solve the System of Equations:

   • Substitute $P(B) = \left(\frac{5}{6}\right) \cdot P(A)$ into $P(A) + P(B) = 1$: $P(A) + \left(\frac{5}{6}\right) P(A) = 1$
   • Simplify: $P(A) \left(1 + \frac{5}{6}\right) = 1$ $P(A) \cdot \frac{11}{6} = 1$ $P(A) = \frac{6}{11}$

Geometric Series Approach

1. Probability Representation:

   • The probability that Amy wins on her first roll is $\frac{1}{6}$.
   • The probability that Amy wins on her third roll is: $\left(\frac{5}{6}\right)^2 \cdot \frac{1}{6}$
   • The probability that Amy wins on her fifth roll is: $\left(\frac{5}{6}\right)^4 \cdot \frac{1}{6}$

2. Infinite Geometric Series:

   • The pattern continues, forming a geometric series with first term $a_1 = \frac{1}{6}$ and common ratio $r = \left(\frac{5}{6}\right)^2$.
   • The sum of the infinite geometric series is given by: $S = \frac{a_1}{1 - r} = \frac{\frac{1}{6}}{1 - \left(\frac{5}{6}\right)^2}$
   • Simplify: $S = \frac{\frac{1}{6}}{\frac{11}{36}} = \frac{6}{11}$

Conclusion

Both the linear equations and geometric series approaches yield the same result: the probability that Amy wins is $\frac{6}{11}$. This solution highlights the importance of understanding probability transitions and the use of infinite series to solve complex probability problems.