Lucky Lily and Winning William decide to play the following...

Lucky Lily and Winning William decide to play the following game based on an urn containing nine white balls and eleven black ones. The play proceeds as follows. A ball is drawn and replaced. If it is white, Lili wins a dollar from Winning. If the ball is black, Winning wins a dollar from Lili. Lili starts with 20 dollars and Winning with 10 dollars. The game continues until one player wins all of the other player's money. What is the probability that Winning wins?

Lucky Lily and Winning William decide to play the following game based on an urn containing nine white balls and eleven black ones.
The play proceeds as follows. A ball is drawn and replaced. If it is white, Lili wins a dollar from Winning. If the ball is black, Winning wins a dollar from Lili. Lili starts with 20 dollars and Winning with 10 dollars. The game continues until one player wins all of the other player's money. What is the probability that Winning wins?

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

Transition Probabilities

These are the probabilities of moving from one state to another in a stochastic process. In games like this, they are determined by the probability of specific outcomes (such as drawing a particular colored ball) and are essential for setting up the equations that describe the evolution of the players' stakes over time.

Random Walk with Drift

A random walk with drift is used to model situations where there is a bias towards one direction due to unequal probabilities of winning or losing a bet. It explains how a systematic drift in the players’ fortunes over time influences the likelihood of one player winning all the money.

Gambler's Ruin

This concept involves determining the probability that one player ends up bankrupt (or wins all the money) in a series of bets where players win or lose money based on chance. It typically uses recurrence relations or closed-form solutions to calculate the outcome probabilities based on the initial stakes of each player.

Markov Chains

This framework models random processes where the future state depends only on the present state and not on the sequence of events that preceded it. In the context of gambling games, states representing the players' fortunes change with fixed probabilities and the process continues until an absorbing state is reached.

Transcript

Please give Ace some feedback