Question

In the gambler's ruin problem of Section 4.5.1, suppose the gambler's fortune is presently \(i\), and suppose that we know that the gambler's fortune will eventually reach \(N\) (before it goes to 0 ). Given this information, show that the probability he wins the next gamble is $$ \begin{array}{ll} \frac{p\left[1-(q / p)^{i+1}\right]}{1-(q / p)^{i}}, & \text { if } p \neq \frac{1}{2} \\ \frac{i+1}{2 i}, & \text { if } p=\frac{1}{2} \end{array} $$

Short answer

The probability of the gambler winning the next bet, given that they will eventually reach N dollars, is: $$ \begin{array}{ll} \frac{p\left[1-(q / p)^{i+1}\right]}{1-(q / p)^{i}}, & \text{ if } p \neq \frac{1}{2} \\ \frac{i+1}{2i}, & \text{ if } p = \frac{1}{2} \end{array} $$

Step-by-step solution

Expression for winning the next bet

If the gambler wins the next bet, they now have i+1 dollars. The probability of them reaching N dollars given that they win the next bet is the same as the probability of reaching N starting with i+1 dollars, which can be represented as P(N|i+1).

Expression for losing the next bet

If the gambler loses the next bet, they now have i-1 dollars. The probability of them reaching N dollars given that they lose the next bet is the same as the probability of reaching N starting with i-1 dollars, which can be represented as P(N|i-1). #Step 2: Use conditional probability formula to find the probability of winning the next bet given reaching N dollars#

Conditional probability formula

Let A be the event of winning the next bet, and B be the event of reaching N dollars. We want to find P(A|B). Using the conditional probability formula, we have: $$ P(A|B)=\frac{P(A \cap B)}{P(B)} $$ We can rewrite the numerator and denominator of this fraction using the expressions from Step 1. $$ P(A|B)=\frac{p \cdot P(N|i+1)}{p \cdot P(N|i+1) + q \cdot P(N|i-1)} $$

Plug in the formula for reaching N dollars

The formula for the probability of reaching N dollars starting with i dollars (found in Section 4.5.1) is: $$ P(N|i) = \begin{cases} \frac{1-(q/p)^i}{1-(q/p)^N}, & \text{ if } p \neq \frac{1}{2} \\ \frac{i}{N}, & \text { if } p = \frac{1}{2} \end{cases} $$ We need to plug this formula into the conditional probability expression from above. For \( p \neq \frac{1}{2} \): $$ P(A|B) = \frac{p\left[ \frac{1-(q/p)^{i+1}}{1-(q/p)^N}\right]}{p\left[\frac{1-(q/p)^{i+1}}{1-(q/p)^N}\right] + q\left[\frac{1-(q/p)^{i-1}}{1-(q/p)^N}\right]} $$

Simplify the expression

We simplify the expression by canceling common terms in the numerator and denominator to obtain the final probability expression for \( p \neq \frac{1}{2} \): $$ P(A|B)=\frac{p\left[1-(q / p)^{i+1}\right]}{1-(q / p)^{i}} $$ For \( p = \frac{1}{2} \), similar steps yield the final probability expression: $$ P(A|B)=\frac{i+1}{2 i} $$ Thus, the probability of the gambler winning the next bet given that they will eventually reach N dollars is: $$ \begin{array}{ll} \frac{p\left[1-(q / p)^{i+1}\right]}{1-(q / p)^{i}}, & \text{ if } p \neq \frac{1}{2} \\ \frac{i+1}{2i}, & \text{ if } p = \frac{1}{2} \end{array} $$

Key concepts

Conditional Probability

Conditional probability is a fundamental concept in statistics that deals with assessing the likelihood of an event occurring given that another event has already happened. This principle is crucial for understanding complex probability scenarios, as it is often necessary to calculate probabilities in a chained sequence of events where prior outcomes influence subsequent ones.

In the context of the gambler's ruin problem, conditional probability allows us to evaluate the chances of the gambler winning the next gamble under the condition that they will eventually reach a certain wealth level, denoted as 'N'. This can be mathematically expressed using the formula for conditional probability:
\[\begin{equation}P(A|B) = \frac{P(A \cap B)}{P(B)}\end{equation}\]
Where A represents the event of winning the next bet and B represents the event of reaching the wealth level 'N'. By calculating the intersection of A and B, and dividing by the probability of B, we can quantify the gambler's chances in a much more precise way. Grasping this concept ensures a better understanding of how events relate to one another within the scope of probability theory.

Probability Formulas

Probability formulas are mathematical equations used to calculate the likelihood of varying events. In many cases, especially in games of chance and stochastic processes, these formulas are essential to predict outcomes and make strategic decisions. For the gambler's ruin problem, we use specific probability formulas to determine the chances of the gambler reaching a certain level of wealth.

The steps detailed in the solution involve applying probability formulas to account for different scenarios: winning and losing a bet. By plugging the results into the conditional probability equation, the gambler's next move can be assessed with greater accuracy. Here's the tailored probability formula for the gambler's scenario, distinguishing between cases based on the gambler's odds ( p ) of winning each individual bet:
\[\begin{equation}P(N|i) = \begin{cases}\frac{1-(q/p)^i}{1-(q/p)^N}, & \text{ if } p eq \frac{1}{2} \frac{i}{N}, & \text { if } p = \frac{1}{2}\end{cases}\end{equation}\]
This equation computes the probability of the gambler achieving a wealth level of 'N' starting with 'i' currency units. It's a pivotal tool contributing to the understanding of long-term prospects in gambling and similar stochastic situations.

Stochastic Processes

Stochastic processes are mathematical models used to describe systems or phenomena that evolve over time in a way that involves randomness or unpredictability. They play a vital role in fields such as physics, finance, and of course, games of chance like those found in casinos.

In studying the gambler's ruin problem, we're dealing with a discrete-time stochastic process. Each bet represents a step in time where the gambler's fortune changes in a random fashion—either increasing or decreasing—with defined probabilities. The process continues until a certain absorbing state is reached, in this case, the gambler's wealth hitting zero or reaching 'N' dollars.

This exercise examines the gambler's evolution based on a sequence of independent and identically distributed random variables, which is characteristic of many stochastic processes. Understanding these processes is key to quantifying risk and making informed decisions in uncertain environments, reflecting the inherent volatility present in systems incorporating random events.