Using Combinations To Calculate Probabilities In Pineapple Open Face

Difficult endgame decisions in Pineapple Open Face Chinese (POFC) are quite common, and it can be quite a challenge to choose the play that offers the highest expected value. The most difficult aspect of calculating these expected values are determining the underlying probabilities of making certain hands.

There are solvers and trainers available for POFC where you can input a particular situation and it will produce the expected values, however, they may not exist for all variations. In addition, if you disagree on a key assumption, such as the value of Fantasyland (FL) that is built into the model, the outputs will not be exactly what you desire.

A working knowledge of combinations is most helpful in calculating probabilities in POFC, and is usually easier than any other approach. When we talk of combinations, we are counting the number of ways in which a group of elements can combine into smaller groups where the order doesn’t matter.

In poker, the order doesn’t tend to matter because for example, A 5 is clearly the same hand as 5 A.

The general combination formula is C(n, k) = (n!) / [(n-k)], where n is the total population and k is the amount of elements in the smaller group. The exclamation point is a factorial, which is a product of all positive integers less than or equal to n, k, or (n-k).

For example, there are four aces in a standard fifty-two card deck and the number of possible combinations when choosing two of them would be:

C(4,2) = (4*3*2*1) / [(2*1)(21)] = 24/4 = 6

If we wish to calculate the probability of being dealt pocket aces, we would divide the result above by the total amount of 1,326 possible hold’em hands:

C(4,2)/C(52,2) = 6/1326 = .45%

If you are working in Microsoft Excel you can use the COMBIN function to quickly calculate combinations.

Let’s look at a few examples to illustrate:

One-Card Draws

Suppose we are playing POFC High and heading into the fourth and final pull we have the following hand:

A-2
3-3-6-6-8
K-K-Q-Q

We are up against an opponent who is in Fantasyland (FL), thus his holding is hidden to us. If along the way we have discarded a deuce, a nine, and a ten, what are the odds that on the final pull we end up making a pair of aces up top and get to FL?

This is a relatively easy problem from a probability perspective as we just need to calculate the chances that we don’t make a pair of aces and then subtract that result from one. There are 38 unseen cards, thus the probability of making aces is as follows:

1. [(35/38)(34/37)(33/36)] = 22.4%

This is also an easy problem using combinations:

1. [ C(35,3)/C(38,3)] = 1- [6,545/8346] = 22.4%

Combinations also allow us to more easily calculate this probability in a direct fashion; when doing that we should calculate the number of combinations involved with making three aces, two aces, and one ace, and add them together for the numerator:

Combinations of Three Aces: C(3,3) = 1
Combinations of Two Aces: C(3,2)C(35,1) = 105
Combinations of One Ace: C(3,1)C(35,2) = 598
Total Number of Combinations for Denominator: C(38,3) = 8,346
Probability of Getting at Least One Ace = [1 + 105 + 1,785]/8,346 = 22.4%

The result will be different when our opponent is not in FL since we have information regarding 17 more cards. With position there are only 21 unknown cards, and if villain only shows one ace the probability of making aces is slightly greater:

1. [C(19,3)/C(21,3)] = 1 – [969/1330] = 27.1%

“Simul Draws”

The next logical question to ask is how often do we expect to make both a pair of aces up top and a full house on the bottom?

Here we need to calculate the probability that we get at least one card from Group A (the remaining aces) and at least one card Group B (the remaining kings and queens) simultaneously within the final pull. In POFC, the calculation of the odds of getting what we require from two separate groups is often referred to as a “simul draw.”

The easiest way to calculate this probability is directly using combinations, and in order to do so we need to carefully list out and sum up the combinations in which we get what we need.

Let’s assume once again that our opponent is in Fantasyland, thus his cards are not visible to us. There are 38 cards left in the deck, broken down as follows:

Group A (Aces) = 3
Group B (Kings and Queens) = 4
Group O (All Other Cards) = 31

Summing up the combinations for the different ways in which we can improve both the top and bottom lines:

One Ace, One King or Queen, and One Other Card = C(3,1)C(4,1)C(31,1) = 3*4*31 = 372
Two Aces and One King or Queen = C(3,2)C(4,1) = 34 = 12
One Ace and Two Kings or Queens = C(3,1)C(4,2) = 36 = 18
Total Combinations = 372 + 12 + 18 = 402
Probability of Success = 402/C(38,3) = 402/8,346 = 4.8%

A general formula for “simul draws” may be expressed as the following:

US = Unseen Cards
A = Outs in Group A
B = Outs in Group B
O = Other Cards not in A or B
Sum of Combinations for Numerator = (A)(B)(O) + COMBINB + ACOMBIN
Denominator Combinations = COMBIN

Using the generic formula, it is possible to create a set of reference charts based upon the number of unseen cards and available outs that one can readily access. In the examples above, we simply examined the probabilities of making our hand on the fourth and final pull, however, in practice our final decision point would occur in how to best place the cards we received on the third deal.

We have access to much more information when our opponent is not in Fantasyland. Not only do we know about more cards and how it impacts the odds of making our desired hands, we must also factor in the potential of winning either two of the lines or scooping our opponent when deciding our best course of action. In the next installment we will continue the discussion and look at a few examples where these calculations are put to use in our decision making. ♠

Kevin Haney is a former actuary but left the corporate job to focus on his passions for poker and fitness. The certified personal trainer owned a gym in New Jersey, but has since moved to Las Vegas. He started playing the game back in 2003, and particularly enjoys taking new players interested in mixed games under his wing and quickly making them proficient in all variants. Learn more or just say hello with an email to [email protected].