[@@ todo - do this better, and talk about when and why it matters; you need the right number of people in the
hand, the right implied odds, and then compare to the chance of flopping what you want]

Whenever you decide to play any two cards you have to think "what am I hoping will come"?  eg. what kind of cards can come on the flop that will make me happy, and then what is the chance that it will happen?  For example, when you play AK, any flop with an A or K makes you happy, a flop with a QJ or JT is ok but not great, and a flop with two of your suit (for AKs) is good.  You see already how AKs is better than AKo - there are many more flops that can come that will make you happy.  Conversely with something like A2, you can't hope for much.  Any flop with an Ace is scary because there may be another ace out there.  Pairing your 2 doesn't do much good, since it will always be bottom pair.  You can only hope for a flop like 345 or A2T, obviously these do not come nearly often enough to play.

The same logic is used for drawing hands like 33.  If you don't hit a 3 you can't be happy, so you're only playing that hole to hit a 3.  That means you have about a 12% chance of seing a flop you like, so you need to be getting that "pot percent" to call.  You shouldn't play it heads up to a raise.

Note that when deciding whether to play some hole cards, it's pretty reasonable to just think about the flop, since once the flop comes you'll have to face more betting action and make a decision again.  By calling pre-flop, you have not earned the right to see all 5 board cards, you've just bought the right to see the flop.  You then have to decide whether to continue or not afresh.

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Odds of flopping something
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If you hold a pair, the chance of flopping trips is 1 - (48/50)*(47/49)*(46/48) = 11.75 %

If you hold a suited hand (11 outs), the chance of flopping

exactly a 2 flush = (39*38*37)/(50*49*48) = 46.62 %
exactly a 3 flush = (11*39*38)*3/(50*49*48) = 41.59 %
exactly a 4 flush = (11*10*39)*3/(50*49*48) = 10.94 %
exactly a 5 flush = (11*10*9) /(50*49*48) = 0.84 %

a 3 flush (or better) : 1 - (39/50)*(38/49)*(37/48) = 53.37 %
a 4 flush (or better) : 11.78 %

chance of the 4 flush is about the same as the chance of trips. Of
course then the chance of the 4-flush improving is 35% (9 outs, two
cards to come).

If you hold a connector (8 outs), the chance of flopping

exactly a 2 straight = (42*41*40)/(50*49*48) = 58.57 %
exactly a 3 straight = (8*42*41)*3/(50*49*48) = 35.14 %
exactly a 4 straight = (8*7*42)*3/(50*49*48) = 6.00 %
exactly a 5 straight = (8*7*6) /(50*49*48) = 0.29 %

This is actually not quite right, it's more complicated, but I imagine
this is close. The thing is, when you hit one straight card, you still
have 8 outs to make a longer straight, but what those outs are depend on
which end of the straight you hit. So, to hit a 4-straight, you have 3
ways : two on the bottom, two on the top, or one to each side. So
actually the chance of a 4-straight (or better) is :

(4*4*48)*9/(50*49*48) = 5.88 %

So, playing with just a connector is not so great. With a suited
connector you will hit a 4-straight or 4-flush about 17% of the time.
(you can't just add probabilities because they overlap a bit). If your
connector is at one end of the spectrum (A2,AK) your chances of the
straight are greatly reduced.

What's the chance of hitting a pair on the flop (with no pair in your hand) ?
You have 6 outs

Chance of one pair or better = 1 - (44/50)*(43/49)*(43/48) = 30.8 %

Chance of top pair depends on the value of your cards.  Let's say you have AK, you will always make top pair if you make a pair; similarly if you have 23 you will always make bottom pair if you make a pair.  How about if you have 9T ?  Then it's just the chance of hitting a pair and no overs coming.

What if you have somethine like 89 or 99 ; the chance of having a flop with no overs is just the chance that all 3 cards are a 9 or lower.  That's 8 ranks, so there are 30 cards, taking off your two.  So the chance is :

(30/50)*(29/49)*(28/48) = 0.207 = 20.7 %

So the chance of 99 seing no overs is very poor, and the chance of 89 flopping top pair is very poor indeed, roughly 20% * 30% = 6%

Chance of two pair or better = (6*5*48)*3/(50*49*58) = 3.04 %
("better" includes trips and four of a kind)

So, if you have something like JTs you have a lot of ways to get something on
the flop - [@@ do computer simulation for this exactly]

4-flush : 12%
4-straight : 6%
pair : 30% (that pair is top pair a little less than half the time, 43%)
2-pair or trips : 3%

These overlap a bit, so you can't just add them, but clearly you have a
good chance of hitting something. If you hit nothing at all, fold.

Chance of having an ace :

1 - (48/52)*(47/51) = 14.932%

Chance of someone out of N having an ace :

1 - (1- 0.14932)^N

For N = 5 , that's 55%

The vast majority of the time in Hold'Em you will be dealing with pairs on the flop.  That's why it's so important to play hole cards that will flop good pairs (eg. play high cards), and to know about top pair vs. middle pair vs. bottom pair.  If you flop top pair you don't have to worry about any pair beating you, you do have to worry about your kicker, but that becomes a Bayesian issue.