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I have a case where there are 2 classes and I want to figure out how class 1 effects the passing rate of class 2.

An example case:

Class   Total Student   Passed  Failed
1       20              15      5
2       25              18      7

And we know that 12 students took both classes and 5 of them passed both classes. 2 failed both classes. 2 students passed class 1 but failed class 2. 3 students passed class 2 but failed class 1.

So, what I am asking is what is the passing probability of course 1(P(course 1)) and passing probability of course 1 given course 2(P(course 1|course 2)).

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$$P(pass1|pass2) = \frac{P(pass1 \cap pass2)}{P(pass2)}=\frac{\frac{5}{12}}{\frac{8}{12}}=\frac{5}{8}=0.675$$

For the total population, the passing rate was 3/4 or 0.75. And if you took class 2, but failed it, P(pass1|fail2) is only 0.25.

Interestingly, all of those who did not take class 2, passed class 1

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$$P(pass1|notake2) = \frac{P(pass1 \cap notake2)}{P(notake2)}=\frac{\frac{8}{20}}{\frac{8}{20}}=1$$

So taking class 2 hurt your chances of passing 1, even if you passed 2!

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