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Chapter 4: Problem 56

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 winning \(n\) or losing \(m\). What is the probability that the gambler quits a winner?

Short Answer

Expert verified
The probability that the gambler quits the game as a winner is: \(P(0) = \dfrac{1}{p}\)

Step by step solution

01

Understand the random walk situation

This problem can be considered as a random walk in one dimension, where the gambler starts at position 0 and moves forward (win) or backward (lose) in each step. The game ends when the gambler reaches position \(n\) (winning) or \(-m\) (losing).
02

Set up the equation for probability

Let's denote the probability of the gambler reaching position \(k\) as \(P(k)\). Since the gambler can only win with probability \(p\) and lose with probability \((1-p)\), we can write the equation for the probability of the gambler reaching a given position as: \(P(k) = p \cdot P(k-1) + (1-p) \cdot P(k+1)\) We can rewrite this equation by isolating the term \(P(k-1)\): \(P(k-1) = \dfrac{P(k) - (1-p) \cdot P(k+1)}{p}\)
03

Determine the boundary conditions for the random walk process

In this problem, we have two boundary conditions that must be satisfied. The first is when the gambler reaches the winning threshold (\(n\)), and the second is when the gambler reaches the losing threshold (\(-m\)). We can express these boundary conditions as: \(P(n) = 1\) (probability of winning when reaching \(n\)) \(P(-m) = 0\) (probability of winning when reaching \(-m\))
04

Identify the required probability

We are interested in finding the probability \(P(0)\), which represents the probability that the gambler quits the game as a winner, starting from the initial position.
05

Set up the recurrence relation

From Step 2, we have the recurrence relation: \(P(k-1) = \dfrac{P(k) - (1-p) \cdot P(k+1)}{p}\) We can express \(P(0)\) using this recurrence relation: \(P(0) = \dfrac{P(1) - (1-p) \cdot P(2)}{p}\)
06

Use the boundary conditions to solve the recurrence relation

Using the boundary conditions \(P(n)=1\) and \(P(-m)=0\), we can iteratively substitute the recurrence relation obtained in Step 5. Each time we apply the relation, we will decrease the index of \(P\) by 1, and we will reach the required probability \(P(0)\) after doing this \(m\) times: \(P(0) = \dfrac{P(1) - (1-p) \cdot P(2)}{p} = \dfrac{ \dfrac{P(2) - (1-p) \cdot P(3)}{p} - (1-p)\cdot P(2)}{p} = \dots\) Continue to apply the recurrence relation \(m\) times to reach \(P(0)\): \(P(0) = \dfrac{P(m) - (1-p) \cdot P(m+1)}{p} = \dfrac{1 - (1-p) \cdot 0}{p} = \dfrac{1}{p}\) Thus, the probability that the gambler quits the game as a winner is: \(P(0) = \dfrac{1}{p}\)

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Most popular questions from this chapter

\(M\) balls are initially distributed among \(m\) urns. At each stage one of the balls is selected at random, taken from whichever urn it is in, and then placed, at random, in one of the other \(M-1\) urns. Consider the Markov chain whose state at any time is the vector \(\left(n_{1}, \ldots, n_{m}\right)\) where \(n_{i}\) denotes the number of balls in urn \(i\). Guess at the limiting probabilities for this Markov chain and then verify your guess and show at the same time that the Markov chain is time reversible.

For the Markov chain with states \(1,2,3,4\) whose transition probability matrix \(\mathbf{P}\) is as specified below find \(f_{i 3}\) and \(s_{i 3}\) for \(i=1,2,3\). $$ \mathbf{P}=\left[\begin{array}{llll} 0.4 & 0.2 & 0.1 & 0.3 \\ 0.1 & 0.5 & 0.2 & 0.2 \\ 0.3 & 0.4 & 0.2 & 0.1 \\ 0 & 0 & 0 & 1 \end{array}\right] $$

Let \(Y_{n}\) be the sum of \(n\) independent rolls of a fair die. Find \(\lim _{n \rightarrow \infty} P\left\\{Y_{n}\right.\) is a multiple of 13\(\\}\)

Let \(\pi_{i}\) denote the long-run proportion of time a given irreducible Markov chain is in state \(i\). (a) Explain why \(\pi_{i}\) is also the proportion of transitions that are into state \(i\) as well as being the proportion of transitions that are from state \(i\). (b) \(\pi_{i} P_{i j}\) represents the proportion of transitions that satisfy what property? (c) \(\sum_{i} \pi_{i} P_{i j}\) represent the proportion of transitions that satisfy what property? (d) Using the preceding explain why $$ \pi_{j}=\sum_{i} \pi_{i} P_{i j} $$

It follows from the argument made in Exercise 38 that state \(i\) is null recurrent if it is recurrent and \(\pi_{i}=0 .\) Consider the one-dimensional symmetric random walk of Example \(4.15\). (a) Argue that \(\pi_{i}=\pi_{0}\) for all \(i\). (b) Argue that all states are null recurrent.
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