How can all 3 of these be true? 
*

*Most numbers are composite.

*If you choose a random whole number there is a 50/50 chance that it's even or odd.

*If you take 2 random whole numbers and multiply them there is a 75% chance the result is even and a 25% chance it is odd.
(That is even*even=even, odd*even=even, even*odd=even, and only odd*odd=odd)


How can all 3 of these be true?
 A: Consider a simpler situation: we'll call the two numbers $0$ and $1$ "binary numbers". Then:


*

*Every binary number can be expressed as a product of binary numbers.

*If you choose a random binary number there is a 50/50 chance that it's even or odd.

*If you take 2 random binary numbers and multiply them there is a 75% chance the result is even and a 25% chance it is odd.


Is that surprising?
A: The first point mentioned does not have any relation to the other points. However it may point to the source of you confusion. Without that point your reasoning would seem to be: random numbers are equally likely to be even and odd, but their product is more likely to be even, how is that possible? Well that simply means that picking two numbers and multiplying them is not a good (uniformly) random way to pick a number. (You need to do something to make "(uniformly) random" meaningful, as there does not exist any uniform probability defined on all of the integers; one way out is to pick random "machine integers" (i.e., in some range $[0,N)$) and define machine multiplication (modulo$~N$) as the product.) The fallacy of this reasoning is even more evident in "random numbers$~n$ are equally likely to be even and odd, but $2n$ is always even, how is that possible?".
The $2n$ example shows that not even all numbers are possible as an outcome of certain methods of picking numbers. You seem to have realised that multiplying two numbers is more likely to produce composite numbers than to produce other numbers (primes or $0,1$). Even then, there is no reason to assume that among the composite numbers the probability of finding one of them after multiplication is uniform, and indeed (in any reasonable formalisation of this formulation) it is not. In fact the error of reasoning is already apparent from the fact that finding a non-composite number by a multiplication is not entirely impossible, if you allow picking $0$ or $1$ as one of the numbers to multiply; there is no absolute reason to limit to composite numbers only, and yet composite numbers are more likely to come out.
But even apart from this there is no basis for your reasoning. The following three statements together are far from logically contradictory.


*

*Half of the population is female, the other half male

*The majority of the population is wage earner

*Most wage earners are men


Therefore, even if you had a sound argumentation to arrive at the conclusion that "most composite numbers are even" (and indeed the complementary statement "most non-composite numbers are odd" is quite defendable, although technically it is not equivalent), this should only lead to the reaction "so what?".
A: Because even ignoring primes because they make a small difference, you are saying 2 random numbers multiplied have a 75% chance of being odd. Sure but there are more repeats in multiplications for even numbers. In other words more multiplications result in even numbers, but since those even numbers aren't necessarily unique even numbers it can be explained by odd composite numbers just having fewer multiplications that result in them. For example, 35 = 7*5 and that's it. But 36 = 2*18 = 3*12 = 4*9 = 6^2. Even though there is one number 36 and one number 35, there are many more multiplications leading to 36. So if you randomly pick numbers out of 2, 3, 4, 5, 6, 7, 9, 12, and 18, and multiply, you are going to get 36 a lot more than you get 35. (You will get 36 7/81 of the time and 35 2/81 of the time.) That's why your statements are consistent, because your 75% probability is for the multiplication result. That doesn't mean more numbers necessarily are even, it can instead mean that even numbers are likely to be more often the result of multiplication because they have more factors. It would instead be strange if 75% of all multiplications with unique results gave even numbers. Then the answer may be that the difference was being caused by odd primes, but obviously that's not the case with our number system.
A: You're question #3 almost answer themselves, if there are 4 possible options, and 3 out of 4 of the possible options give us an even product, than there is a 75% chance the product is even and a 25% chance that the product is odd. The first 2 questions are general proporties which can be shown by taking any 2 consecutive numbers as examples
