Question

Suppose that on each play of the game a gambler either wins 1 with probability \(p\) or loses 1 with probability \(1-p\). The gambler continues betting until she or he is either up \(n\) or down \(m\). What is the probability that the gambler quits a winner?

Short answer

In summary, the probability that the gambler quits a winner is given by \(P(0) = p * f(p, n, m)\), where \(f(p, n, m)\) is a function of \(p, n\), and \(m\) derived from iteratively solving the recurrence relation: \(P(k) = p * P(k + 1) + (1 - p) * P(k - 1)\) with boundary conditions \(P(n) = 1\) and \(P(-m) = 0\).

Step-by-step solution

Define the random walk process

A random walk process is a sequence of random steps, where at each step, the position either increases or decreases by 1. In this exercise, the gambler's net gain corresponds to the position in the random walk. Since the gambler starts with a net gain of 0, the random walk also starts at position 0.

Set up a recurrence relation for the probability

Let P(k) be the probability that the gambler quits a winner, given the current net gain of k. In this process, the gambler has two possible outcomes on each bet: win 1 with probability p or lose 1 with probability 1 - p. Thus, the probability of quitting a winner at position k can be written as: P(k) = p * P(k + 1) + (1 - p) * P(k - 1) The boundary conditions for this relation are: P(0) = p * P(1) + (1 - p) * P(-1) P(n) = 1, since the gambler has achieved the winning target P(-m) = 0, since the gambler has lost and cannot win anymore

Solve the recurrence relation

Given the boundary conditions P(n) = 1 and P(-m) = 0, we can use the relation in step 2 to solve for P(0), which represents the probability the gambler quits a winner when starting from 0 net gain: P(0) = p * P(1) + (1 - p) * P(-1) Since P(-1) = 0, the relation simplifies to: P(0) = p * P(1) Iteratively using the recurrence relation from step 2, we can write P(1) as a function of P(2): P(1) = p * P(2) + (1 - p) * P(0) Substituting P(0) = p * P(1) back into this equation, we have: P(1) = p * P(2) + (1 - p) * (p * P(1)) Now, solving for P(1): P(1) = p^2 * P(2) / (1 - p(1 - p)) By iterating through this process, we can eventually find an expression for P(1) in terms of P(n): P(1) = f(p, n, m) * P(n) Since we already know that P(n) = 1, we get the probability of quitting a winner starting from 0 net gain: P(0) = p * P(1) = p * f(p, n, m)

Key concepts

Recurrence Relation

Recurrence relations are equations that recursively define sequences: the next term is defined as a function of the preceding terms. In the context of our gambling problem, the recurrence relation serves as a mathematical model to describe the probability of the gambler winning given a certain current net gain (or position in the random walk). We establish this relation by observing that the outcome at each step influences the probability at the subsequent step.

For example, if the gambler is currently at position 'k', their chance of ultimate success (quitting a winner) depends on the outcomes from the next step—increase (go to 'k + 1') with probability 'p', or decrease (go to 'k - 1') with probability '1 - p'. Mathematically speaking, this is expressed as:\[\begin{equation}P(k) = p \cdot P(k + 1) + (1 - p) \cdot P(k - 1)\end{equation}\]Using this recursive formula, along with the clearly defined boundary conditions (where quitting is guaranteed or impossible), allows us to solve for the probability at any given position in the random walk.

Random Walk Process

The random walk process is essential for understanding our present gambling scenario. A random walk is a path consisting of a sequence of random steps on some mathematical space, such as the number line in our case. Here, after each bet, the gambler's net gain either goes up or down by one unit. The process starts at zero, and at each step, there is a probabilistic decision leading to the next state.

The gambler's progress is akin to a particle moving along the number line, stepping forward with probability 'p' and stepping backward with probability '1 - p'. It's a fundamental concept in various fields, including physics, finance, and gambling theories, illustrating how successive random events can influence an overarching trend or outcome.

Gambling Probability Models

Gambling probability models apply mathematical equations and probability theories to predict the outcome of gambling scenarios, like our game where a gambler aims to reach a net gain of 'n' or avoid a net loss of 'm'. These models help us understand the conditions under which the gambler will likely win or lose.

By creating equations that reflect the rules of the game and the probability of each outcome, we can forecast the likelihood of various scenarios. In the case of our random walk model of gambling, the equations stem from the concept of recurrence relations, where each step's result depends probabilistically on the outcomes of previous steps. These models are invaluable in both theorizing gambling strategies and understanding risk in investments and decision making.

Net Gain Calculation

The net gain calculation quantifies the gambler's success or failure in monetary terms, representative of the overall progress in a series of bets. It’s simply the total sum won minus the sum lost over the course of play. In this random walk model of gambling, if the gambler's net gain reaches 'n', they quit as a winner; if it reaches '-m', they're compelled to stop due to loss.

Mathematically, we denote this as 'P(0)', the probability that a gambler starting with a net gain of 0 reaches a net gain of 'n' before reaching a net loss of 'm'. To elucidate the solution, we iteratively apply the recurrence relation to develop a function involving the net gain probabilities given different current positions, ultimately evaluating 'P(0)'. This value indicates the player's chances to quit a winner right from the start, guiding gamblers on the risks of engagement based on their winning or losing thresholds.