You will recall our previous columns on raisin' and chasin' where we raised pre-flop in late position with a modest hand, for example, 6d-5d. In part 6 of this series, we presented a table showing the number of clean outs you need in order to see the turn card. The number of outs was dependent on the number of opponents you could count on to pay to see the turn card also. That table applies to fixed-limit hold 'em where betting limits double on the turn and river. In this column we show the number of outs you need to pay to see the river card. And it, too, depends on the number of active opponents.

We start with the drop. These days the drop usually costs $5: $4 for the rake and $1 for the bad-beat drop.

Either you raised before the flop or you called a raise before the flop. You or someone else bet the flop and the turn. The pot contains five small bets multiplied by the number of active players, minus the drop, plus any dead money from the players who folded. (We reserve the dead money and "implied" odds to compensate for those times when you make your hand and still lose to a better one.)

Your pot odds equal (the number of your opponents times five, minus the drop, plus the three small bets you already put in the pot)-to-2.

We know that winning chasers have pot odds larger than their cards odds. Winning chasers always have money odds larger than their cards odd, while losing players don't.

Your cards odds equal (46 minus the number of your Outs)-to-Outs. We combine that and the pot odds into this formula,

#Opponents >_ (46 / (#Outs) + Drop / 2B) / 2.5 - 1

By #Opponents we mean active opponents who put in five small bets. If an opponents acts after you and might not put in that last bet, then you don't count her.

After the turn, 46 cards remain unknown which we divide by the number of your outs. We divide the drop by two small bets, so for a $2-4 hold 'em game, the drop divided by $4 equals 1.25. For a $10-20 game, it equals 0.25. We divide by 2.5 and then subtract 1 to arrive at the number of active opponents needed for you to pay to see the river, depending on your outs.

Solving that same equation differently, the table shows the minimum number of outs you need to pay to see the river card, depending on the number of opponents upon whom you can count to see the river. You have two active opponents and only six outs? Then don't pay to see the river!

We solved the equation for the popular limits shown. At limits of $8-16 and up the numbers of outs you need don't change.

Notice the "Rake Effect." At $2-4 limits with one opponent, you need 13 outs to pay to see the river card. At stakes of $8-16 and higher with one opponent, you need 10 outs to pay to see the river card.

In summary, with one raise before the flop and one bet after the flop, depending on the stakes at which you play and the number of your opponents, you need the number of clean outs shown to chase profitably. By following the chart (not shown) you won't chase riparian rainbows.

Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold 'Em, on sale at amazon.com & kokopellipress.com. E-mail your Hold 'Em questions to richardburke@comcast.net