Players A and B play a sequence of independent games. Player...

Players $A$ and $B$ play a sequence of independent games. Player $A$ throws a die first and wins on a "six." If he fails, $B$ throws and wins on a "five" or "six." If he fails, $A$ throws and wins on a "four," "five," or "six." And so on. Find the probability of each player winning the sequence.

Key Concepts

Conditional Probability

This concept refers to finding the probability of an event occurring given that another event has already occurred. In sequential games, the outcome of a later trial (or turn) often depends on the fact that all previous trials failed to produce a win. This conditioning on past failures is essential when calculating the chances of winning in any round after the initial one.

Independent Events

In probability theory, events are independent if the outcome of one does not affect the outcome of another. In the context of the sequential die tosses described in the problem, each die throw is an independent event. This property allows us to multiply probabilities when looking at a series of tosses, since the chance of a win in one turn doesn’t influence the chance on another turn.

Sequential Trials

Sequential trials refer to processes in which several attempts or events occur one after the other, with each trial having its own probability of success. In this scenario, players take turns rolling the die, and the winning condition improves with each successive turn. Understanding sequential trials is important for setting up the overall probability calculation across multiple rounds.

Stopping Conditions

A stopping condition is a predetermined rule that ends a sequence of events once it is met. In the game described, a player wins as soon as their roll meets the win condition, thereby stopping further play. Determining the probabilities involves computing the chance that none of the earlier stopping conditions were met (i.e., no one has won) before a particular trial, and then weighting the probability of winning on that trial.

Recursive Probability and Series Summation

This concept involves calculating the overall probability of an outcome (such as a win) by summing the probabilities over all rounds—each conditional on the failures of all previous rounds. Because the win condition changes with each turn, the probability for each round is computed recursively, taking into account the series of previous failures. The final overall winning probability is then obtained by summing these conditional probabilities over all possible rounds.

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