An investor owns shares in a stock whose present value is 25...

An investor owns shares in a stock whose present value is $25 .$ She has decided that she must sell her stock if it goes either down to 10 or up to $40 .$ If each change of price is either up 1 point with probability .55 or down 1 point with probability $.45,$ and the successive changes are independent, what is the probability that the investor retires a winner?

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Key Concepts

Gambler's Ruin

The Gambler's Ruin framework addresses the probability that a process will eventually be absorbed by one of the boundaries, often characterized by a loss of all resources or the achievement of a target. It provides formulas for calculating the chance of reaching a winning or losing boundary based on initial conditions and step probabilities.

Absorbing Boundaries

Absorbing boundaries are specific states in a stochastic process that, once reached, terminate the process. They represent conditions under which the system stops evolving; for example, a process may be absorbed when it reaches a set minimum or maximum value, signifying an end point.

Markov Process

A Markov process is a type of stochastic process characterized by the memoryless property, meaning that the probability of transitioning to any particular future state depends solely on the present state and not on the history of past states. This concept simplifies the analysis of sequential, random events.

Random Walk

A random walk is a stochastic process in which successive movements or steps are made in random directions according to fixed probability rules. It serves as an essential model in various fields, such as finance and physics, for describing the path of a variable that evolves over time through a series of probabilistic changes.

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