A gambler has 100 and makes a series of 1 bets that he can...

A gambler has $\$ 100$ and makes a series of $\$ 1$ bets that he can correctly guess the outcome of a coin flip heads or tails. (The coin and the game are honest, so for every coin flip, there is a $50 \%$ chance that the gambler will guess correctly.) The gambler will quit when he has either won an additional $\$ 100$ or has lost all his money. About how many times will the gambler bet before quitting?

A gambler has $\$ 100$ and makes a series of $\$ 1$ bets that he can correctly guess the outcome of a coin flip
heads or tails. (The coin and the game are honest, so for every coin flip, there is a $50 \%$ chance that the gambler will guess correctly.) The gambler will quit when he has either won an additional $\$ 100$ or has lost all his money. About how many times will the gambler bet before quitting?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Random Walk

A random walk is a mathematical model that describes a path consisting of a succession of random steps. In the context of gambling, each bet can be viewed as a step where the gambler either moves one unit up or one unit down based on the outcome of a coin flip.

Gambler's Ruin Problem

The Gambler's Ruin problem is a classical probability problem that analyzes the likelihood of a gambler either reaching a target wealth or losing everything given a series of bets with certain win/loss probabilities. It serves as a framework for understanding how chance influences the eventual outcome of repeated bets.

Absorbing Boundaries

Absorbing boundaries are fixed limits in a random walk model that, once reached, stop the process. In problems like the Gambler's Ruin, the gambler stops betting when reaching either a predetermined wealth or zero, making these boundaries 'absorbing' states where the random process terminates.

Expected Time to Absorption

The expected time to absorption refers to the average number of steps or bets taken before the random walk hits one of its absorbing boundaries. This expectation is often computed using formulas that take into account the starting position and the distance to the boundaries, revealing that the duration can increase quadratically with the size of the interval.

Transcript

Please give Ace some feedback