Question

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.

Short answer

The probability of player A winning the sequence is \(\frac{9}{14}\), and the probability of player B winning the sequence is \(\frac{5}{14}\).

Step-by-step solution

Evaluate Probability for Each Turn

Initially, probability of Player A winning is \(\frac{1}{6}\) and Player B winning is \(\frac{2}{6} = \frac{1}{3}\). After A fails, in the second turn, the probability of A winning is \(\frac{3}{6} = \frac{1}{2}\). Remember that the probability of A winning in the second turn includes that A has failed in the previous turn.

Calculate Total Winning Probabilities

Total probability of A winning, P(A), can be summed as a geometric series: \(P(A)=\frac{1}{6}+\frac{5}{6}*\frac{1}{3}*\frac{1}{2}+\frac{5}{6}*\frac{2}{3}*\frac{5}{6}*\frac{1}{3}*\frac{1}{2}+...\). Here, the first term provides the probability of A winning in the first turn, while every subsequent term follows the pattern of A failing, then B failing, and ultimately A winning. Applying geometric series formula \(S = a / (1-r)\), we find \(P(A) =\frac{1}{6} / (1-\frac{5}{9})\).

Solve for B’s Winning Probability

Since A and B are the only two players, and probability of either A or B winning is 1, so probability of B winning, P(B), can be calculated as \(P(B) = 1 - P(A)\)

Compute Final Probabilities

Substituting probability of A winning from step 2 to step 3, we calculate the final probabilities: \(P(A) = \frac{9}{14}\) and \(P(B) = \frac{5}{14}\).

Key concepts

Probability Theory

Probability theory is a branch of mathematics concerned with the analysis of random phenomena. The central objects of probability theory are random variables, events, and their probabilities. As seen in the provided exercise, the foundation of probability theory allows us to calculate the likelihood of certain outcomes when actions'such as rolling a die'have inherently random results.

In our example, players A and B are taking turns in rolling a die, with specific conditions laid out for each player to win. Probability theory helps us figure out the chance of these events taking place. By understanding probability distributions, which describe the likelihood of different outcomes, we can apply mathematical formulas to evaluate each player's chances of winning across multiple turns.

Keep in mind that in probability, all outcomes must be accounted for, which leads to an important principle: The sum of probabilities of all possible outcomes in a given scenario equals one. This is why when we calculate player A's chances of winning, the leftover probability is attributed to player B, since these two events cover the entirety of possible outcomes.

Geometric Series in Probability

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This series is incredibly useful in probability, especially when dealing with independent events that occur in succession.

In the exercise, when player A does not win on the first roll, they get a second chance with increased odds. This situation repeats, creating a sequence of events that can be expressed as a geometric series. The series constructed in our probability problem captures the essence of this type of scenario: a series of opportunities for an event (in this case, player A winning) with adjusted probabilities as the game continues.

Mathematically, the sum of a geometric series can be expressed with the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term of the series, and \( r \) is the common ratio. Applying this formula simplifies the process of finding the total probability by summing infinitely many terms, as we utilize it to find player A's total winning probability in the example.

Probability Problem Solving

Problem solving in probability involves identifying the right methods to apply to the given situation. In the example provided, recognizing that the sequence of events follows a geometric series allows us to approach the problem using the formula for the sum of a geometric series.

Key practice steps for solving probability problems include identifying all possible outcomes, their probabilities, and understanding how these outcomes are related, whether they are independent or dependent events. Calculating each step can be a methodical process. We incorporate prior events (such as player A failing to win in the first round) into the calculations of subsequent rounds.

Moreover, improving clarity in probability problem-solving can be achieved by breaking down complex sequences into understandable components. It's essential to keep in mind how prior events impact the probability of future events, which is elegantly dealt with by the geometric progression and underpins the iterative nature of calculating probabilities in such multi-stage games, as seen in the exercise solution.