River odds

Let's start by using the flush hand that we've looked at before. You're holding A8, the flop comes K65.

You know that there are 9 remaining spades = 9 flush outs. Say the J comes on the turn, and now we're at the river. Now of the 52 total cards in the deck we know 6 so there are 52 - 6 = 46 remaining unseen cards. 46 unseen cards less the 9 spades = 37 times you will not get a spade. Your odds are 37 to 9. Divide 37 by 9 and you get 4.1 to 1 odds of hitting the flush on the river. Note that this is only marginally lower than on the turn because the situation has only changed by one card.

Because the situation has only changed marginally in terms of known/unknown cards, the odds of hitting one of your outs on the river as opposed to the turn is slightly better on the river than the turn, but not enough that it is going to make a significant difference in your situation.

Considering the Turn and River together

Here we get into an area that is often misunderstood by beginners. We'll go back to our flush draw hand. You've got A8, the flop comes K65.

There are 9 outs to fill the flush. You know 9 outs gives you 4.2 odds of hitting the flush on the turn. If you miss, you will have 4.1 odds of hitting the flush on the river. But on the turn, with both the turn and river cards to come, your odds must be better than 4.2 to get one spade on the turn and/or the river.

Step-by-step, you have 52-5=47 unseen cards on the turn. 9 are spades, so 47-9=38 non-spades. On the river there are 46 cards and 9 spades remaining. 9/46*38 = 7.4 times the non-spade turn will be followed by a spade. Add those 7.4 to the 9 where you hit the spade on the turn, 9 + 7.4 = 16.4 times a spade hits out of 47 times on the turn so 47 - 16.4 = 30.6 to 16.4 or 1.86 to one odds.

So, rounded, the odds are 1.9 to 1 against getting a spade on either the turn or on the river.

1.9 to 1 are pretty good odds, so now you can call that flop bet, right? Not necessarily. Preflop you called the $10 blind, the small blind folded his $5 and the big blind checked his $10. $25 in the pot when the big blind bets $10 into you, giving you $35 to $10 to call pot odds, or 3.5 to 1. If you look at this and see the 3.5 to 1 pot odds as better than your 1.9 to 1 flush draw odds, you're missing the fact that you will likely have to call another bet on the turn if you miss the flush on the turn.

Say you call the $10 turn bet and the turn is the J. The pot is $45 to start, and the big blind bets $20. Now you have to put in $20 to win $45, or 45 to 20 or 2.5 to 1 odds. There is only the river card to come, and you know that your chance of hitting the flush on the river is 4.1 to 1 so now you fold. In a strict application of pot odds theory and assuming that you need the flush to win, you should have folded on the turn but you were misled by neglecting to consider that you may have to call a turn bet.

From your opponent's perspective, this is why you need to bet the turn as well as the flop, to charge your opponents for making the mistake of trying to draw to beat your hand if, in fact, they are chasing the draw. If on the other hand you happened to have a King with a better kicker than your opponent then your opponent is building the pot for you by trying to make draws pay, but neither of you know that for sure.