Poker decisions reduce to comparing two percentages. The first is how often your hand improves to a winner. The second is how often it needs to, given the price. Everything else is reading people.
Counting the first number
An out is a card that makes your hand. Holding four cards to a flush after the flop, there are nine remaining cards of that suit, so nine outs.
The standard shortcut is the rule of 4 and 2: multiply outs by 4 on the flop, when two cards are still to come, and by 2 on the turn, when one is. Nine outs gives roughly 36 per cent on the flop and 18 on the turn, which the poker odds calculator reports along with a warning worth heeding.
The warning is that the rule of 4 drifts. The exact figure for nine outs on the flop is 34.97 per cent, so 36 is a little generous. At twelve outs the rule says 48 and the truth is 44.96. At fifteen it says 60 against a real 54.12. The approximation double-counts the overlap between hitting on the turn and hitting on the river, and the overlap grows with the number of outs.
The rule of 2 errs the other way, slightly. Nine outs on the turn is exactly 9 in 46, which is 19.57 per cent, so the rule understates by about half a point. Below eight outs both are close enough to use at the table. Above twelve, take a few points off the flop number.
Counting the second number
Pot odds are the price. If the pot holds 100 and your opponent bets 50, calling costs you 50 to win a final pot of 200. You break even when you win often enough to recover that:
required equity = call ÷ (pot + bet + call)
Fifty divided by two hundred is 25 per cent. That is the threshold. Win more often than a quarter of the time and calling makes money, however the individual hand turns out.
Putting them together
Flush draw, pot 100, bet 50. On the flop your equity is about 35 per cent against a required 25, so calling is correct by a clear margin.
Now the turn comes blank and your opponent bets 50 into 200. You need 50 divided by 300, which is 16.7 per cent, and you have 19.57. Still a call, though thinner.
Change the turn bet to 150 into 200 and the required equity becomes 150 divided by 500, which is 30 per cent. Your 19.57 is nowhere near it, and the same hand that was an easy call one street ago is now a clear fold. Nothing about the cards changed. The price did.
This is the entire skill in the mechanical part of the game: your equity falls as streets pass because fewer cards remain, and opponents raise the price as the pot grows. A draw is a call early and a fold late at the same nominal bet size.
Two things the ratio leaves out
Implied odds are money you expect to win on later streets when you hit. They justify calls that pot odds alone do not, and they are the most abused concept in poker because the amount is always a guess, usually a generous one.
The other is that outs are not always clean. Nine flush outs are only nine if the flush wins when it arrives, and against a board that could pair into a full house some of those cards lose. Count outs that win, not outs that improve.
For the raw frequency of each hand in a five-card deal there is the poker probability calculator, where a flush comes in at 0.197 per cent and a full house at 0.144. To read a specific five cards, there is poker hand strength. And for the same arithmetic without the money attached, see the piece on drop rates.