Hold'em Odds

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Hey people, how's it going...

So I've heard and read (just on this board) conflicting information on hold'em odds and how to calculate them. I'm looking for some solid information or at least a discussion of which way is probably the right way. What I'm mainly concerned with is decision making post-flop in NL games. Ya know, do I have the odds to call an X sized bet and chase my flush draw, etc. etc.

Here's how I learned it, and this may be wrong... you can tell me. Btw, I'm most comfortable with percentages rather than odds since I don't gamble outside of poker and percentages is what I learned in math class.

Post-Flop betting:
Chances you'll make your hand on the turn: outs/47
Chances you'll make your hand on the turn or river: This is the one that is confusing, so I'll discuss it below.

Post-Turn betting:
Chances you'll make your hand on the river: outs/46

Okay... Chances you'll make your hand on the turn OR river. I have read a variety of poker odds web sites and looked at a number of odds calculators, and there are definitely different ways people are calculating the odds for this. My instinct is to go with math, and not to trust sites that base odds on however many simulated hands. I mean, why go with a simulation when I can calculate precisely how often that third Ace I need is going to come up?

This is my understanding of the statistics behind poker odds:
If you want to know the chances of events X AND Y happening, like catching a runner-runner flush, you multiply the chances of each happening individually. Here, it would be (10/47)*(9/46), which is 4.17%.

What I don't get exactly is how to calculate the chances of event X OR event Y happening. For example, what are the odds of completing an inside straight draw by the river? The chances of catching it on the turn by itself is (4/47), and the chances of catching it on the river if you missed on the turn is (4/46). How do you put those together? You don't multiply. Adding doesn't seem to make sense. I saw one site that calculated the odds of you NOT getting the cards you need, since that can be an "AND" situation and you can multiply. It was like this:

Odds you'll complete your inside straight draw:

Start with odds you won't hit on turn AND won't hit on river: (43/47)*(42/46) = 83.53%

Subtract this number from 100% to find out chances that this WON'T happen (i.e. chances you'll have you straight by the river): 100% - 83.53% = 16.47%

Is this correct? Is there another way? I thought this was the right way, but I'll show one last example where someone else's numbers contradicted mine.

In the MTT forum, there was an all-in situation where it was basically

A[img]/images/graemlins/spade.gif[/img] A[img]/images/graemlins/club.gif[/img] against 3[img]/images/graemlins/spade.gif[/img] 3[img]/images/graemlins/club.gif[/img]

The flop was: 2[img]/images/graemlins/diamond.gif[/img] 4[img]/images/graemlins/heart.gif[/img] 5[img]/images/graemlins/diamond.gif[/img]

In the thread, the guy with As said he had about a 75% chance to win and backed it up with numbers from http://www.twodimes.net/poker/. But the guy with 3s had only four outs (the sixes, since the other two threes would give As a straight) or he would lose. According to the statistics as above, that means he only had a 16.47% chance to catch a winning card, making the guy with As about a 83% favorite. I say about because I get that if both a 3 and a 6 came then they would tie with a straight to 6.

So... okay, I've laid out my thought process and some examples. Tell me why I'm right or why I'm wrong, and thanks very much for doing so.