Increasing probability

Apr 2010
40
1
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

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Aug 2008
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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.
 
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Apr 2010
40
1
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