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There is one formula in math that is the most important formula in Bridge. And that formula is the “Combination Formula”. So, every Bridge player should know, respect and understand the absolute true nature of this formula for Bridge.

The formula tells you how many combinations that are possible if you chose a certain number of items out of a collection. For example, I have a collection of 5 cards in my hand: A, K, Q, J and T (Ten). So, how many 2-cards combinations possible to get from that collection? We can figure it out manually. Here they are AK, AQ, AJ, AT, KQ, KJ, KT, QJ, QT and JT. So, 10 possible combinations. And it is impossible to get another combination, right?

Thus, this formula can tell straight away the number without threading it one by one like above. Just a bit of simple math calculation. You will see later that this formula will be very useful and handy for playing better Bridge.

The Most Important Math Formula in Bridge: an Introduction

Below is the formula in its official math notation. Do not fear the appearance because you will understand it within the few next paragraphs. Promise!

The Combination Formula
The combination formula, the most important math formula in Bridge

In this formula:

  • C() is just a symbol of a formula or function. In this case a combination formula.
  • n = the total number of item in the collection. For example, the collection of 5 cards above make it n=5.
  • r = combination of items that you want to pick. For example, r=2 when you want to pick 2-cards combination from the collection mentioned above.
  • ! = factorial operation. ‘Factorial’ is just a fancy math term to describe multiplication from 1 to that very number. For example:
    • 2! = 2 x 1 = 2
    • 5! = 5 x 4 x 3 x 2 x 1 = 120
    • etc.

So, with our example above, let’s calculate the total combinations with the formula. So the total combinations of picking 2-cards combo (r=2) from a set of 5-cards above (n=5) will make:

Calculating combination

And there you have it. It calculated exactly 10 combinations as we have already know from example above.

How is this important formula useful in Bridge?

In general, the formula will be useful to offer a Bridge player a “percentage play”. The kind of line of play that give the best possible result in the long run. The line of play that agree with the statistics. The statistics favours this play!

For example, an analysis what we call “Outstanding Cards” play -or simply “Missing Card” play.

So, here is a typical scenario. You play a trump contract where you can see 5-cards trump in your hand and another 4-cards on the dummy. Meaning, your side has a total of 9-cards. There are 4-missing-cards -or- 4-outstanding-cards that opponents have.

Therefore, to avoid one trump looser (if you can), since you are missing the Queen, will you:

  • play “drop” (playing A and K in the hope to drop the Queen)? -Or-
  • finesse the Queen (a 50 – 50 chance if without any info from the bidding)?

Thus, the most important formula in Bridge will make it possible for you to have this familiar table to use:

SplitPercentage
4 – 09.6%
3 – 149.7%
2 – 240.7%

So, with the knowledge from the table above, you know that playing “drop” only work 41%. In this way finesse with a 50% chance is better. Or you can make it even better. For example to cash the Ace first just in case a singleton queen in 3-1 split occurred.

Outstanding Cards Detail Analysis

To make it even better, you should be familiar with “Outstanding Cards Detail Analysis” (will be available soon, wait for the link here) that will give you a more specific/detail table as follows:
(LHO= Left-Hand Opponent, RHO = Right -Hand Opponent)

SplitPercentageLHORHOSpecific PctRef
4-04.8%Qxxx(void)4.8%(1)
3-124.9%Qxxx18.7%(2)
xxxQ6.2%(3)
2-240.7%Qxxx20.3%(4)
xxQx20.3%(5)
1-324.9%Qxxx6.2%(6)
xQxx18.7%(7)
0-44.8%(void)Qxxx4.8%(8)

With the above “Specific Outstanding Card Table” you know your chance even better. In this case, playing Ace and then finesse will only fail at (5), (7) and (8). The failure odd: 20.3% + 18.7% + 4.8% = 43.8% , meaning the odds of success is 56.2%. As a result, with almost 15% additional chances, it becomes a significant improvement compared to just dropping the Queen that only has 40.7% odd.

Just a reminder about “Percentage Play”: it will definitely work in the long run. Individual event will give you a random result. But just like the casino make millions knowing the odds are with them, they do not worry about individual loss, but aggregately, you will get better result playing Bridge with the odd on your side.

Know your odds, start play a better Bridge!

So, hopefully, that is a good introduction to the most important formula in Bridge. And, if you want to know even more how this formula produces such a useful table, (hopefully not the only thing that a math science gives you practical application in real-world) you can click the link about The Outstanding Cards Statistic in Bridge and Outstanding Cards Detail Analysis on their corresponding link.

In conclusion, if you know your statistics, getting better result is just a matter of time…

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Math in Bridge