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Let's say we have a parameter $r$ and a binary event $A$ repeatedly happens. The event is binary, so the outcome is either $0$ or $1$. We have collected a lot of data of the form $\{\{r_1,A_1\},\{r_2,A_2\},\cdots,\{r_n,A_n\}\}$ where $r_i\in\mathbb{R}$ and $A_i\in\{0,1\}$.

For example: $\{\{-3,0\},\{-2,1\},\{2,1\},\{2,1\},\{1,0\}\}$

Can we somehow estimate the probability of $A$ being $1$ for a certain $r$. From the example data, it seems when $r=2$ that $A=1$ quite certainly. But the data sample is very very large and I'm totally at a loss at how to estimate this probability. When there are a lot of positive outcomes for certain values of $r$ than that increases the probability of a positive outcome for other values close to $r$.

How can all this be accumulated in order to predict (and how confidently) the probability of a positive outcome once we set an arbitrary $r$?

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    In your example the only time $r = 1$ gives a value of $0$ for $A$.2011-10-06
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    oops, that's $r=2$, gonna fix that.2011-10-06
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    @vedran Do you assume $r$ follows a continuous distribution ?2011-10-06
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    No, $r$ is a totally random real number. Say, the average grade in high-school. And $A$ would be e.g. "person graduated". Then you collect all people and get their average grade, and whether they eventually graduated. I ask then: Based on that data, with my average grade 4.23, how likely am I to graduate?2011-10-06
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    @vedran How dig of data-set are you talking ?2011-10-06
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    About 130000 $\{r,A\}$-pairs.2011-10-06

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