Two persons A and B, each roll a pair of dice, alternately...

Two persons $\mathrm{A}$ and $\mathrm{B}$, each roll a pair of dice, alternately till one of them gets a sum, equal to 4 or 7 and wins the game. If A starts the game the probability that he wins the game is
(a) $\frac{2}{7}$
(b) $\frac{4}{7}$
(c) $\frac{3}{7}$
(d) $\frac{5}{7}$

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Key Concepts

Dice Outcome Probabilities

This concept deals with calculating probabilities for various outcomes when rolling dice. It involves understanding the sample space for two dice, determining the probability of specific events (such as sums of 4 or 7), and using these probabilities as building blocks for analyzing more complex scenarios in games. Mastery of dice probability helps in quantifying the chance of success or failure for a given roll.

Recursive Probability Analysis

This concept involves setting up and solving recurrence relations to model ongoing processes, such as a game where players repeatedly take turns until a win occurs. Recursive probability analysis captures the idea that the game resets with the same probabilistic structure after each pair of turns, allowing the overall winning probability to be expressed as a sum of probabilities over an infinite sequence of trials.

Turn-Based Game Dynamics

This concept covers the strategy and probability analysis in games where players alternate turns. It involves understanding how the starting player's order can impact the overall probability of winning, and requires careful evaluation of opportunities and risks at each turn. The analysis often builds on first-step analysis, where outcomes of the initial move shape subsequent play and cumulative winning chances.

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