In a game two players A and B take turns in throwing a pair...

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of 7 before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is : [Main Sep. 04, 2020 (II)] (a) $\frac{5}{31}$ (b) $\frac{31}{61}$ (c) $\frac{5}{6}$ (d) $\frac{30}{61}$

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of 6 before $B$ throws a total of 7 and $B$ wins the game if he throws a total of 7 before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is :
[Main Sep. 04, 2020 (II)]
(a) $\frac{5}{31}$
(b) $\frac{31}{61}$
(c) $\frac{5}{6}$
(d) $\frac{30}{61}$

Key Concepts

Markov Chain and Absorbing States

A Markov chain is a stochastic process where the next state depends only on the current state, not the history. In many game scenarios, certain states are 'absorbing', meaning that once entered, the process stops. Understanding these absorbing states, which represent winning or terminating conditions in a game, is crucial for calculating overall probabilities in sequential settings.

Probability of Dice Outcomes

This concept involves calculating the likelihood of each possible result when rolling a pair of dice. It requires understanding the complete sample space of dice rolls and how to count favorable outcomes for specific events using combinatorics. It forms the basis for determining individual event probabilities in many games and random processes.

Conditional Probability

Conditional probability is used to evaluate the likelihood of an event occurring given that another event has occurred. In sequential games, it helps in updating the probability of subsequent events based on previous outcomes. This is particularly important when outcomes on one player's turn can affect the overall probability of winning the game.

Sequential Decision Processes

Sequential decision processes involve a series of steps or turns where the outcome of one step influences future decisions or probabilities. This concept is central in turn-based games where players alternate actions and the game outcome depends on the sequence of moves, requiring the analysis of interdependent events.

Turn Order Advantage

Turn order advantage refers to the impact that the sequence of play has on the probability of winning in a game. Analyzing who goes first can be critical when success in a game is determined by achieving a particular outcome before the opponent does. This concept highlights how initial conditions or positions can create asymmetries in competitive situations.

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