Here are the basic rules of probability, from which everything else can be figured out:

- Equivalent properties have the same probability
- Probabilities are nonnegative
- Anything guaranteed to be true has probability 1
- If A1, A2, A3, …, are such that any two of them are incompatible, then P(A1 OR A2 OR A3 OR …) = P(A1) + P(A2) + P(A3) + … [This is the addition rule]
- P(A AND B) = P(A) * P(B | A), where the second term means “The probability of B given that A is true”. [This is the conditional probability rule]

We can derive some other things from these rules. For example, 1 = P(A OR NOT A) by rule 2, and this in turn is equal to P(A) + P(NOT A) by rule 3. So, P(NOT A) = 1 - P(A). [This is what I’m guessing you mean by the subtraction rule]

What’s P(A OR B) in general? Well, A OR B is logically equivalent to (A AND B) OR (A AND NOT B) OR (B AND NOT A), all of which are pairwise incompatible, so by rules 0 and 3, P(A OR B) = P(A AND B) + P(A AND NOT B) + P(B AND NOT A). This in turn is equivalent to P(A) + P(B) - P(A AND B), since A is equivalent to (A AND B) OR (A AND NOT B) and the “mirror-image” equivalence holds for B. [This is the inclusion-exclusion rule **ultrafilter** mentions]

So those are some nice rules. But what about P(A AND B)? Can we figure it out just from knowing P(A) and P(B)?

No, in general, we cannot. Just knowing P(A) and P(B) doesn’t tell us anything about the relationship between A and B; they might be incompatible, or they might be equivalent, or they might be independent, or any number of other things. We need to know which of these is the case in order to figure out P(A AND B). [By definition, A and B are independent if P(B | A) = P(B); i.e., being given B as true doesn’t change the probability of A].

If A and B *are* independent, then, by rule 4, we have that P(A AND B) = P(A) * P(B). [This is the multiplication rule]

So… to answer your specific questions:

P(female AND blonde) = P(female) * P(blonde | female) = p(blonde) * p(female | blonde). *If* we know that being female and being blonde are independent traits, then this equals P(female) * P(blonde).

P(female OR blonde) = P(female) + P(blonde) - P(female AND blonde)

P(dying your hair blonde 100 or more times during your lifetime among females) = whatever it is. It’s not something we can derive automatically from P(blonde) and P(female). Conceivably, if you have a particular probabilistic model for hair-dying in mind, we could come up with some formula for it in terms of other, simpler quantities, but you haven’t given any such model.