Stochastic Simulation: The Gambler

Consider the following simulation. A gambler starts with an initial bankroll (e.g., $50) and a target goal (e.g., $100). He or she then makes a series of $1 bets (winning each with independent probability 0 ≤ p ≤ 1) until reaching the goal or going broke.

We are interested in simulating this experiment to approximate two results:

• The chance of the gambler reaching the goal.
• The average number of bets until the simulation ends.

To do so, we wish to have you implement a function, documented as follows.

function [success bets] = gambler(goal, initial, p)
   % Performs independent trials for a gambling simulation.
   % USAGE: [success round] = gambler(goal, initial, p)
   %   A gambler begins with a bankroll designated as initial
   %   and repeatedly bets $1 with a winning percentage of p
   %   continuing until either meeting the goal or going broke.
   %
   %   If not otherwise specified,
   %       initial = goal/2
   %       p = 0.5

Challenge for Class

Implement the gambler function, as parameterized above. Then write separate scripts to estimate the answer to each of the following (using a minimum of 1000 trials).

• For a goal of $100 and an initial bankroll of $50, what is the probability of success if the gambler has a 49% chance of winning each individual bet?
• How many bets, on average, will be made for the above simulation?
• For a goal of $100 and a probability of 49%, what is the minimum initial bankroll needed to provide at least a 50% chance of achieve the goal?

In the long run, we will use this gambler function to perform broader analysis where we allow one or more of the parameters to vary. For example, here is a graph showing the experiment done with a fixed goal and p-value but varying initial bankroll.

This shows that with an initial bankroll of 5, the chance of success is only 23% or so. To have the chance of success near 50%, given the p-value, we would need an initial bankroll of 7.

To do these tests, modify your function to perform a series of trials, as such:

function [success bets] = gambler(goal, initial, p, trials)
   % Performs independent trials for a gambling simulation.
   % USAGE: [success round] = gambler(goal, initial, p, trials)
   %   A gambler begins with a bankroll designated as initial
   %   and repeatedly bets $1 with a winning percentage of p
   %   continuing until either meeting the goal or going broke.
   %
   %   success will be the fraction of times in which goal was reached
   %   bets will be the average number of bets placed before the outcome
   %
   %   If not otherwise specified,
   %       initial = goal/2
   %       p = 0.5
   %       trials = 1