Historically, interest in stochastic processes (and probability in general) came from gambling. The following is a classic process that people have analysed (and which will look at in great detail during the first half of the course). It models the wealth of a gambler after successive bets. People analysed this process to try and see if there was any strategy of gambling which would be certain to make money.
Start at , and plot for = 1, 2, 3 …, corresponding to successive flips of the coin, an increase of by 1 for a Head and a decrease of by 1 for a Tail; can be thought of as the accumulated wealth of a gambler at time .
STOP if becomes 0. We suppose it stays at that value thereafter.
The experiment is best done in pairs. One of the pair flips a coin repeatedly, the other draws a graph of vs . Then swap over.
At the end, count for the whole class, the total number of graphs terminating at = 1, 3, 5, 7, …, including those still going on beyond, say, 30. We indicate these by * to indicate the possibility that there is no finite time at which these would terminate.
| 1 | 3 | 5 | 7 | 9 | 11 | [13,30] | * | |
From these we can estimate the probabilities of terminating at these time points.