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Math Help - Increasing probability

  1. #1
    Junior Member
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    Apr 2010
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    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
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  2. #2
    MHF Contributor

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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 (N_k)_{k\geq 0} in [0,1], you can define a process (X_n)_{n\geq 1} with values in \{0,1\} such that, given X_1,\ldots,X_n, the probability of X_{n+1}=1 is N_{K(n)} where K(n)=\#\{1\leq i\leq n: X_i=1\}. If (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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  3. #3
    Junior Member
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    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
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