This document provides lessons on percentage plays in bridge, focusing on situations where a certain number of cards are missing key honors like aces, kings, queens, etc. It discusses the statistical probabilities of different distributions of the missing cards and the best plays to maximize tricks based on the numbers. Examples are given for various scenarios involving 8-10 cards where different plays like cashing honors, leading low to finesse, or finessing twice are analyzed based on the potential outcomes and probabilities. Safety and avoiding giving opponents ruffs is emphasized over unnecessary finessing.

1. Bridge Lessons - IntermediateNYU Bridge and Spades Club

2. Lesson 14Percentage Plays

3. General IntroductionYou’ve probably been told before that Bridge is a game of statistics. That is absolutely correct and any player who wants to become a good bridge player has to have a basic understanding of which line of play will yield the best results (probabilistically).However, do note that the percentage plays detailed in this lesson assume that the opponents have given you no signals and did not bid at all. Opponents’ bids and signals can give you information on 1. which key cards they hold and 2. their distributions. These will obviously affect which line of play you should choose.

4. Some Statistics for Those who are InterestedP(Opp has A of B missing cards) = BCA*26-BC13-A / 26C13P(Opp has 3 of 5 missing cards) = 5C3*21C10 / 26C13 ~ 0.339 i.e. P(5 missing cards break 3-2) = 2*0.339 = 0.678P(Opp has 2 of 4 missing cards) = 4C2*22C11/26C13 ~ 0.407 i.e. P(4 missing cards break 2-2) = 0.407P(Opp has 1 of 4 missing cards) = 4C1*22C12/26C13 ~ 0.249 i.e. P(4 missing cards break 1-3) = 2*0.249 = 0.498

5. Some Statistics for Those who are InterestedIn general, an even number of missing cards tends to break unevenly while an odd number of missing cards tends to break the most even way possible.The probabilities change according to the number of cards you know in each opponent’s hand.

6. Some Statistics for Those who are InterestedFor example, let us consider the case when you are playing in a spades contract with West having opened 1H earlier. That shows that West has 5 hearts i.e. you already know that 5 of West’s cards cannot be trumps. Assume you and your partner are holding 9 trumps. Now how do the 4 remaining trumps tend to break?

7. “Rule” of 8 Ever, 9 Never“Eight ever- nine never” is a saying that says whether you should finesse a missing queen when you have 8 and when you have 9 cards in the suit. With just 8 cards you should always finesse - eight ever; but with nine cards you should play for the drop - nine never (finesse). But actually the odds for nine cards are very close and you may prefer to finesse if there is an inference that that may be working.

8. Some Statistics for Those who are InterestedP(West has 0 trumps) = 4C0*17C8/21C8 = ~ 0.119 (Why 21C8? This is because you can simply exclude 5 hearts from the remaining 26 cards and pick 8 from the last 21 to obtain all of West’s possible hands that are of concern to you)P(West has 1 trump) = 4C1*17C7/21C8 ~ 0.382P(West has 2 trumps) = 4C2*17C6/21C8 ~ 0.365P(West has 3 trumps) = 4C3*17C5/21C8 ~ 0.122P(West has 4 trumps) = 4C4*17C4/21C8 ~ 0.0117

9. Some Statistics for Those who are InterestedThus, west probably holds 1 trump and it is usually right to finesse through east (assuming you hold A in your hand as south and K in the dummy, you play a low to the K and finesse through east) if missing the Q even though the rule 8 ever, 9 never tells you not to! This is a very simplistic explanation as you will come to learn that there are many other considerations when at the bridge table but is a good foundation to have.

10. 10 Cards Missing KYou are playing NS. N holds AJTxx, S holds Qxxxx. You need all the tricks.Percentage play is to lead Q from S and finesse the K.You only lose when E is holding a singleton K. Note that if E had Kx or Kxx, you would lose to the K either way.When W has singleton K, it doesn’t matter whether you intended to finesse or not.You gain when W has Kx, Kxx or Kxxx. Thus, this is obviously superior to all other lines of play.

11. 10 Cards Missing K – An ExtensionNow what if we tweak this a bit and N doesn’t have the T trump?The finesse must specifically be leading Q from South. In the first case, it didn’t matter you started off leading the Q or low to the JT. In this case, however, should you lead low to the J and end up discovering that W holds KTx or KTxx, you will end up having no way to finesse West for both anymore.

12. 9 Cards Missing JYou are playing NS. N holds A9xxx and S holds KQTx. You need all the tricks.You must lead low to K first. If you discover that one of your opponents started void, you can still finesse for the J. If you had led a low to the A on the first trick and W turns up void, you’ll have no way left to finesse the J.

13. 9 Cards Missing JTSupposed A is dealt with Axxxx and S has KQ98. Should your play be any different from before?The answer is, rather surprisingly, yes. Consider the case when you lead low to K first. If W started with JTxx, you may try to lead 9 from S to N, but all W has to do is play a T on top of it and you will eventually lose to the J. By the same logic, E holding JTxx will result in you losing.However, if you were to play A first, you can still lead low to KQ98 and finesse the JT out. You can’t do anything about W having JTxx, but at least you gain when E has JTxx.

14. 9 Cards Missing QJNorth has AKT while S has xxxxxx. You need all the tricks.It would be wrong to start out by finessing with T directly. You will lose whenever E is holding Qx, Jx, Q or J, only gaining when W is holding QJx (you don’t even gain when W is holding QJxx… figure out why).Start by cashing the A. If both follow small to this trick, cash the K (9 never). However, if E follows with Q or J, you have the choice to finesse. By the mathematical principle of restricted choice, you should return to S in another suit and finesse with the T (contradicting 9 never). The difference between cashing K here and finessing is very small (~2%) and your play will generally be determined by hints at the table/in the bidding.

15. 9 Cards Missing KJN holds AQTxxx, S holds xxx. You can afford to lose 1 trick.This looks like a delicious hand with which to finesse. However, the correct play is to cash the A on the first round. Then cross back to S in another suit and lead low.This loses when W was dealt KJxx or KJx. But remember that you can afford to lose 1 trick and we want to play safety first, especially in trump contracts.If W and E started with 2 each, you will lose 1 trick no matter what you do. If E started with KJx or KJxx, you will lose 2 tricks no matter what you do.Now what if E started with K or KJ? Finessing first will cause you to lose a trick. Then on your second trick, you have yet another decision to make – to finesse or not to finesse for the other honour. Rather than sweat over this decision, it’s better to just cash A on the first round.

16. 9 Cards Missing KQN has AJTxx, S has xxxx. You can afford to lose 1 trick.It’s quite clear here that you should finesse twice, hoping honours are split or are both on W.You only lose when E is holding KQ. If E had KQx or KQxx, there was nothing you could’ve done.The alternative plan of cashing A directly is quite terrible. It only gains when E has KQ but loses when E has x or void. The probability of E holding KQ specifically is <7%.

17. 9 Cards Missing AQN has KJT9xx, S has xxx. You can afford to lose 1 trick.The right play is to lead low to the J.Let’s start by thinking about the possible scenarios. First consider when opponents have 2-2 break. If W has Ax, you should play K. If W has Qx, you should play J. These cancel each other out.For 3-1, playing K works only if E were dealt with singleton Q. However, it loses when E was dealt with any other singleton. Playing J also works better when E started void.You can’t do anything about E holding AQx or AQxx.

18. 8 Cards Missing KJN holds Axxx, S holds QT98. You can afford to lose 1 trick.The best play here is to lead T from S. It doesn’t really make a difference whether you lead Q or T from S in most cases. However, in the case when E has KJxxx, leading T will allow you to recover – you can cover T with A and lead low to Q98 to lose only 1 trick in this suit. Of course, if this suit were the trump suit, you’d go down. This only helps in NT games.

19. 8 Cards Missing QJN holds AKT9xx, S holds xx. You can afford to lose 1 trick.If E started with QJxx or QJxxx, there’s nothing you can do. So let’s worry only about W holding QJxx.The right play here is to play A first, then lead low from S to finesse. Assuming no honours drop when A is played, finessing or cashing K will result in the same number of tricks lost (1) if the break is 3-2. However, if W started with QJxx, cashing AK loses 2 tricks. Thus, you should finesse on the second round.Finessing on the first round loses if E started with a singleton Q or J.

20. 8 Cards Missing QTN has Axxx, S has KJ98. You can afford to lose 1 trick.If you could not afford to lose any tricks, you would have to cash A and lead low to J (8 ever). However, now you just need to avoid losing 2 tricks.Counter-intuitively, you should cash K first then lead 9. If E can win the 2nd trick, the last honour will fall under the A and you have guaranteed 3 tricks.This play wins when W is dealt QTxx. If W follows low twice, you can let the 9 run and win the 2nd trick. Then you’ll have guaranteed 3 tricks as well.This play also wins then E is dealt QTxx. W will show out on the 2nd round and you can take with the A. Then you’ll have J8 riding on top of QT, and that will definitely take 1 trick.No other method offers the same guarantee this method of play does.

21. Notes Before We ConcludeMany budding bridge players make the terrible mistake of finessing whenever they can to take more tricks. Don’t be one of them!Sometimes, finessing runs the risk of opponents getting ruffs. If that is a possibility, try to avoid finessing if you can make the contract without finessing.Always remember, safety first!

22. Congratulations!You are now an Intermediate Bridge player!