Part: Part Four - Pot Odds & Hand Analysis

Counting outs and the price

Pre-flop
Blinds 100 / 200Pot 900 (4.5 BB)BTNA9 20,000 (100 BB)YOUto actMP 20,000 (100 BB)Solid MPraises 600 (3 BB)SB 20,000 (100 BB)posts 100 (0.5 BB)BB 20,000 (100 BB)posts 200 (1 BB)D

A solid player raises from middle position and it folds to you on the button with A♠9♠ - a suited ace that plays well in position.

Solid MP raises to 600 (3 BB); you hold A♠9♠ on the button. Best?

WhyCall. A suited ace flops flushes, straights, and top pairs, and you have position. Calling keeps the pot manageable and sets you up to use the math on later streets.
What happensYou call; the blinds fold. Heads-up in position.  Pot: 1,500 (7.5 BB).
Flop
Heads-upPot 1,500 (7.5 BB)K72BTNA9 20,000 (100 BB)YOUnut flush draw (9 outs)MP 20,000 (100 BB)Solid MPbets 800 (4 BB)D

Flop: K♠ 7♠ 2♦ - you flop the nut flush draw, nine outs. He continuation-bets.

Nut flush draw (9 outs). He bets 800, so you call 800 to win 2,300 (~2.9-to-1). Best?

WhyCall. Nine outs is about 35% to make the flush by the river (and ~19% on the very next card). You're laid roughly 2.9-to-1, which needs only ~26% - so the price alone justifies it, and the implied odds (your nut flush will often win a big pot) make it easy.
What happensYou call.  Pot: 3,100 (15.5 BB).
Turn
Heads-upPot 3,100 (15.5 BB)K724BTNA9 20,000 (100 BB)YOUthe nut flushMP 20,000 (100 BB)Solid MPbets 1,500 (7.5 BB)D

Turn: 4♠ - you make the nut flush. He bets again.

You hit the nut flush and he bets 1,500 (7.5 BB). Best?

WhyRaise for value. You have the nuts; raise to get value from a king, two pair, a set, or a worse flush. This is the reward for the math you trusted on the flop.
What happensYou raise to 4,500; he calls with K-Q.  Pot: 12,100 (60.5 BB).
River
Heads-upPot 12,100 (60.5 BB)K7248BTNA9 20,000 (100 BB)YOUthe nut flushMP 20,000 (100 BB)Solid MPchecksD

River: 8♦ - a blank. He checks.

Nut flush, he checks the river. Best?

WhyBet for value. He still has plenty of hands (a king, two pair) that pay a final bet. Size it to get called - checking the nuts gives up the last bet.
What happensYou bet 6,000; he calls.  You win a big pot.
You counted nine outs, turned them into ~35% equity, compared that to the 2.9-to-1 price, and called - then got paid in full when the flush came in. A draw is a math problem, and the math said go.

Count outs, convert them to equity, and call when the pot odds (plus implied odds) beat the price.