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Apologies in advance if the answer to this question seems obvious to some of the readers but I am confused as to how I should proceed when interpreting the following data.

I have a sample of 5000 students (Male (M) / Female (F)) and I have calculated the probabilities of students passing or failing Test 2 given how they performed in Test 1 (Pass / Fail denoted as subscripts in the tables below). The number of occurrences is also included as "frequency"

TABLE 1 - In row 1, I randomly select either Male or Female / Pass or Fail Test 1 and the resulting probability of passing or failing Test 2 is shown. In row 2, I select another candidate and add it to the pre-existing condition in row 1. I continue this sequence adding onto already previous results up to row 5.

TABLE 2 - I follow the same random selection and the first 2 rows yield the same result as Table 1 giving the same probability of students passing test 2. However conditions diverge in row 3, row 4, row 5, yielding different probabilities.

My question I guess is whether there is a method to intutitively understand the data or some methodology to adjust / standardize the probabilities to reflect that differing frequencies and the addition of conditions to the data set thus making some probabilites more significant than others.

Thank you for your help.

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Instead of the tables you could do a logistic regression where the dependent variable is binary (pass/fail on test 2). The predictor variables would be gender and outcome on test 1. The model will estimate the probability of success on test 2 given the predictor values. If you get additional data you can incorporate it and get new estimates. Initially you should check the goodness of fit of the model to make sure that the estimates are reasonable.

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    $\begingroup$ Michael, initially I thought I´d have to apply some sort of statistic and find some threshold value to make sense of the probabilities but your answer has directed towards logistic/probit regressions. I very much appreciate your guidance as using this type of regression seems more intuitive and is precisely what I was looking for. Thanks for the help. $\endgroup$ – David May 27 '17 at 11:38

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