# Increasing probability

#### salohcin

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

#### Laurent

MHF Hall of Honor
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.

• salohcin