- #1

hjam24

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- TL;DR Summary
- Initially we have a probability of someone winning a game with certain scoring rules. The probability is of winning depends on the probability of winning a point, p, (which is assumed to be constant). The goal is to draw p from a beta distribution and change the formula accordingly

Assume that players A and B play a match where the probability that A will win each point is

If we know that A wins, his score is specified by B's score; he has necessarily scored

In the case of

$$ P(A_{wins} \cap y|p) = \binom{10 + 10}{10}p^{10}(1-p)^{10}

\cdot[2p(1-p)]^{y-10}\cdot p^ 2$$

The elements represents respectively:

- probability of reaching

- probability of reaching

- probability of A winning two times in a row

I would like to change the constant

The first part can be rewritten as as [beta-binomial](https://en.wikipedia.org/wiki/Beta-binomial_distribution) function:

$$ P(A_{wins} \cap y|\alpha, \beta) =\binom{10+10}{10}\frac{B(10+\alpha, 10+\beta)}{B(\alpha, \beta)} \cdot \space _{...} \cdot \space _{...}$$

The original formula can be simplified to

$$P(A_{wins} \cap y|p) = \binom{10 + 10}{10}p^{12}(1-p)^{10}

\cdot[2p(1-p)]^{y-10}$$

Is it correct to combine the first and third element as follows:

$$ P(A_{wins} \cap y|\alpha, \beta) =\binom{10+10}{10}\frac{B(12+\alpha, 10+\beta)}{B(\alpha, \beta)} \cdot \space _{...} $$

*p*, for B its*1-p*and a player wins when he reach*11*points by a margin of*>= 2*The outcome of the match is specified by $$P(y|p, A_{wins})$$If we know that A wins, his score is specified by B's score; he has necessarily scored

*max(11, y + 2)*pointsIn the case of

*y >= 10*we have$$ P(A_{wins} \cap y|p) = \binom{10 + 10}{10}p^{10}(1-p)^{10}

\cdot[2p(1-p)]^{y-10}\cdot p^ 2$$

The elements represents respectively:

- probability of reaching

*(10, 10)*- probability of reaching

*y*after*(10, 10)*- probability of A winning two times in a row

I would like to change the constant

*p*assumption and draw*p*from a beta distribution.The first part can be rewritten as as [beta-binomial](https://en.wikipedia.org/wiki/Beta-binomial_distribution) function:

$$ P(A_{wins} \cap y|\alpha, \beta) =\binom{10+10}{10}\frac{B(10+\alpha, 10+\beta)}{B(\alpha, \beta)} \cdot \space _{...} \cdot \space _{...}$$

The original formula can be simplified to

$$P(A_{wins} \cap y|p) = \binom{10 + 10}{10}p^{12}(1-p)^{10}

\cdot[2p(1-p)]^{y-10}$$

Is it correct to combine the first and third element as follows:

$$ P(A_{wins} \cap y|\alpha, \beta) =\binom{10+10}{10}\frac{B(12+\alpha, 10+\beta)}{B(\alpha, \beta)} \cdot \space _{...} $$