What is the probability of success when two independent predictors co-occur? Let's say I have a collection of objects.  Each object has a set of predictive attributes where the attribute may be true or false.  Each attribute when true predicts "success" with a known probability, and these probabilities are not necessarily the same for different attributes.  What is the probability of success for an object with more than one true attribute?  For example, if being male has a 25% chance of success, and being under 35 has a 50% chance of success, what are the chances of success for a male under 35, assuming the two attributes are independent in terms of success?
Second question.  Let's say I know the z-value for attribute A, and the z-value for attribute B.  What would I expect the z-value to be for (A and B), assuming A and B are independent?
Thanks.
 A: If two events are independent, then $P(A|B)=P(A)$. If on the other hand, two independent drawings are made from two urns to determine whether you find at least one red ball, then the probability becomes $1-(1-(P(\not R_A))*(1-P(\not R_B))$ if $R_A$ corresponds to "drawing a red ball from urn A".
It is not straightforward to link this with yout sentence: "assuming the two attributes are independent in terms of success". 
Given the numbers you have shown the first interpretation is not easy t apply as there are three "events": some one is over 35 (A), someone is male (B) and someone is successful (C).
Independen could mean: $P(C|A) = P(C|A \land B)$ and $P(C|B) = P(C|A \land B)$. But this only works if $P(C|A)=P(C|B)$, which is not the case in the example.
The second interpretation would be equivalent to saying: You basically have two indepndent shots at being successful, one is based on your age, one on yor gender (this makes of course not much sense conceptually or empirically, but would correspond to the run case). In this case I would calculate: $P(C)= $1-(1-(P(C|A))(1-P(C|B))=1-((1-0.25)(1-0.5))=1-0.375=0.625.
The answer by Hotaka uses yet another idea of independence that uses a sequential process: only for those candidates that are not successful based on the first criterion, is the chance for being successful based on the second criterion the specified probability. I would not consider this to be a valid case of independence.
Regarding the second question, I am a bit lost as to how binary attributes are linked to z-values here...
A: 1st question: 
Think about it this way: consider one factor at a time. First, 25% of males will succeed. So no need to worry about them anymore, let's worry about the 75% that are left behind. As long as the factors are independent, 50% of them will succeed. So 75% x 50% = 32.5% will succeed. Finally, add 32.5 and 25 and you get 57.5% success rate.
2nd question is interesting. Imagine a scatterplot of z scores for A on the y-axis and B on the x-axis. They are independent, so basically you'd see a circle with a lot of dots in the middle and fewer as you go outward. The z score of A and B combined is the distance between that specific dot of interest and the very center of the circle, which is 0. So let's go back to the Pythagorean theorem:
z of A and B = ((z of A)^2 + (z of B)^2)^.5
