Increasing probability

Hello all,

suppose there is a probability x that an event happens (and clearly 1-x it doesn't). The catch is, x is not constant, it depends on how many times x has happened before. In other words, with a probability x, the event happens and x increases. With a probability 1-x, the event doesn't happen, and x remains the same. Hence the probability of the event happening increases in how many times the event has happened.

I know this isn't a lot of detail. I am just looking for a suggestion of how to begin modelling something like this. I don't know where to even start. Does anybody have any quick comments about where I should look?

Thank you very much,

Nick

This is called a reinforced process. Simplest example is Polya's urn: Consider an urn with balls of two different colors. Each time we pick a ball, we put it back together with a new ball of the same color (hence this colors becomes more likely).

What you're asking for is quite vague. For any sequence of numbers $\displaystyle (N_k)_{k\geq 0}$ in $\displaystyle [0,1]$, you can define a process $\displaystyle (X_n)_{n\geq 1}$ with values in $\displaystyle \{0,1\}$ such that, given $\displaystyle X_1,\ldots,X_n$, the probability of $\displaystyle X_{n+1}=1$ is $\displaystyle N_{K(n)}$ where $\displaystyle K(n)=\#\{1\leq i\leq n: X_i=1\}$. If $\displaystyle (N_k)_k$ is increasing, this is a general reinforced process. The choice of this sequence is the only data you need.

Thanks for your response,

Basically every time the event occurs, the probability of it happening again increases by x%. For example, suppose the probability of drawing a red ball is .5, then if you draw a red ball today the probability you draw a red ball tomorrow is .55 (if x=.1) etc... I want to express as a function of time t, the expected number of times the event has happened at any given time. Is this possible to do? Thanks in advance,

Nick