| n1 | n2 | n3 | p |
|---|---|---|---|
| 1 | 0 | 1 | ? |
| 0 | 1 | 1 | ? |
| 0 | 0 | 0 | ? |
If I know the variable's n1, n2 and n3 as either true or false (1/0). And for each variable I know that, given its probability, what the probability of p is. How would I calculate the probability of p considering all the variables?
For example, for
| n1 | n2 | n3 | p |
|---|---|---|---|
| 0.8 | 0.2 | 0.6 | ? |
then if n1 is 1, the probability of p is 0.8. How would I combine these together mathematically?
If you simply want to find the probability of having X positive results in Y number of trials, you can use joint probabilities of each situation.
Looking at just the first trial, the probability you get a positive result from $n_1$ is $p_1$. What you can do is for getting one true result in the first trial:
$P(one)_1$= $p_1$ * (1-$p_2$) * (1-$p_3$) Or $P(one)_2$ = (1-$p_1$) * $p_2$ * (1-$p_3$) Or $P(one)_3$= (1-$p_1$) * (1-$p_2$) * $p_3$
In this case shown above, you consider every possible way you can get a single positive result. You use the negations of the non-positive results as for there to be a single positive, the rest must not be. From here your net probability of getting a single positive result is:
$P(one)_1 + P(one)_2 + P(one)_3$
Because you have very few variables in your example, manually calculating as I have done is a viable method. If you have a larger data set, you can still use this method but coding a summation function of each probability combination would make it significantly easier.
I hope this helps.