Can I determine the statistical significance of conditional probabilities? I have the following table of conditional probabilities.
       | pre work | positive outcome based on pre work type 
 group | done     | multi choice       | symbolic           
-------|----------|--------------------|--------------------
 A     | YES      | 0.51               | 0.33               
 B     | YES      | 0.42               | 0.56               

Looking at the table above, is there any statistical test that I can use to confirm/reject that subjects in groups A who have done pre work did better preparing with symbolic vs multi choice work?
(it looks like the hypothesis is rejected, but the question holds anyway)
 A: You should also know the size of each group to estimate the error of the estimates of your conditional probabilities. If in the A-YES-symbolic group only 1 of 3 people succeeded, the estimate 0.33 of the success rate is less reliable than if it's 1000 of 3000. If you don't have that information, I don't think any statistical test would apply.
Update: ok, you have the information, perfect. The rest is just Z-test for independent proportions. Fill the following table with your data:
       |               |  # of outcomes 
 group | pre work type | positive | total           
-------|---------------|----------|-------
 A     | multi choice  | k_1      | n_1               
       | symbolic      | k_2      | n_2               

Let $p_1$ be the probability of positive outcome in multi choice group, $p_2$ — in symbolic group. To test $H_0\colon p_1=p_2$, calculate $$Z = \frac{\hat{p}_1-\hat{p}_2}{\sqrt{P\left(1-P\right)\left(\frac1{n_1} + \frac1{n_2}\right)}}, $$
$$\hat{p}_1 = \frac{k_1}{n_1}, \;\hat{p}_2=\frac{k_2}{n_2}, \; P = \frac{\hat{p}_1n_1+\hat{p}_2n_2}{n_1+n_2}.$$
Under $H_0$ $Z$ would be distributed as $N\left(0,1\right)$, so, if you have $H_1\colon p_1\neq p_2$, then your p-value is $p=2\left(1-\Phi\left(\left|Z\right|\right)\right)$, where $\Phi$ is a standard normal cumulative distribution function.
It might be a good idea to calculate confidence interval for the difference between probabilities as well, so you would have an estimate of the effect size with explicitly specified uncertainty.
