Using Combinations To Calculate Probabilities In Pineapple Open Face: Part 2

In the last installment we discussed combinations and how they may be used to calculate probabilities in Pineapple Open Face Chinese (POFC). Most notably these probabilities are useful when making endgame decisions, those that occur after receiving your third deal of three cards.

A common POFC calculation is the “simul draw,” which is the probability of improving multiple lines by pulling from different groups. For example, in my last POFC column we calculated the probability of making both a pair of aces up top and a full house on the bottom on our final three-card deal when holding:

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

Out of the final three cards we will receive we need one of the three remaining aces (Group A), and also one of the four remaining kings or queens (Group B).

However, we should not confuse “simul draws” with the times when we need multiple cards from the same group. One example of this is when we have a flush draw and require that two of the three cards from the fourth and final pull are of a particular suit. We can’t use the generic “simul draw” formula presented in the last installment, and should just look to calculate from first principles.

Flush Draws

Suppose our opponent is in Fantasyland and our hand is the following:

A 2 A
9 5 5 3 9
K Q 8 X X

Suppose our discards are the 4, 6, and 8. What are the odds that we qualify…by making a flush, trips, or two pair on the bottom? It’s impossible to make both a flush and trips/two pair, so there is no potential for double counting. Therefore, we can just calculate the probability for each of the components and then sum them up.

Let’s start by calculating the probability of making a flush. There are 38 unseen cards and the full complement of 10 spades left in the deck.

Combinations of Three Spades: C(10,3) = 120
Combinations of Two Spades: C(10,2)C(28,1) = 4528 = 1260
Probability of a Flush = (120+1260)/C(38,3) = 1380/8436 = 16.4%

Now let’s move onto the probability of making trips or two pair. All of the kings and queens are live, but one of the eights is dead.

The different ways we can end up with trips are as follows:

Three Kings = C(3,3) = 1
Three Queens = C(3,3) = 1
Three Eights = Not Possible = 0
Two Kings and any other card = C(3,2)C(35,1) = 335 = 105
Two Queens and any other card = C(3,2)C(35,1) = 335 = 105
Two Eights and any other card = C(2,2)C(36,1) = 136 = 36
Total Combinations of Trips = 1+1+0+105+105+36 = 248
Probability of Trips = 248/8,346 = 3.0%

Finally, we can move onto the chances of making two pair. In these calculations we must be very careful not to double-count. For example, when tabulating the number of combinations that make us two pair we can’t include the combinations that make us trips since we already counted those. Also for the combinations of kings and eights, the third card cannot be a queen. Yes, this can get quite complicated!

Keeping this in mind, the combinations for a possible two pair are as follows:
King, Queen and any other card other than another King or Queen = C(3,1)C(3,1)C(32,1) = 3*3*32 = 288
King, Eight and any other card other than another King, Queen or Eight = C(3,1)C(2,1)C(30,1) = 3*2*30 = 180
Queen, Eight and any other card other than another King, Queen or Eight = C(3,1)C(2,1)C(30,1) = 3*2*30 = 180
Total Combinations of Two Pair = 288+180+180 = 648
Probability of Two Pair = 648/8,346 = 7.8%

When it’s all said and done, we have a 27.2% [16.4% + 3.0% + 7.8%] chance of making a qualifying hand.

Suppose that we arrived in this situation because on the third pull we opted to put a pair of aces up top and make two pair in the middle. When our opponent is in Fantasyland it’s a no brainer to make this gamble since he will never foul and if we didn’t are almost certainly getting scooped anyway. In this example our flush outs are fully live, but even if they weren’t it’s correct to throw the Hail Mary.

If some of our spade outs were dead, the corresponding probabilities of making a flush would be as follows:

Flush Outs Probability’
10 16.4%
9 13.4%
8 10.6%
7 8.1%
6 5.9%

When Your Opponent is not in Fantasyland

Much more information is available to you when your opponent’s hand is visible. You can see how strong his board is developing, how likely he is to foul, and more cards are exposed to evaluate how likely our draw would be to come in.

If your opponent is likely to scoop, then it is time to gamble, and if he is likely to foul, then you should play it safe. If it’s somewhere in between, we must consider our upside potential in addition to how live our outs are.

Let’s consider a simplified example; suppose it appears likely that if we choose to play it safe our opponent will not foul, and will win two out of the three lines for a net loss of one point to us. Also, by playing it safe it’s not expected that we will win any bonuses or take a trip to Fantasyland.

However, if we choose to gamble we have the potential to scoop our opponent, get some royalties coming from a pair of aces up top and a flush on the bottom, as well as make it to Fantasyland. In POFC High, a trip to Fantasyland is worth around eight points, which is slightly more than in the Deuce to Seven variation.

If we qualify by making a flush on the bottom, our upside is as follows:

Points
A-A Top 9
Fantasyland 8
Scoop 6
Flush on Bottom 4
27

And if we end up qualifying by making two pair or trips on the bottom our final point tally would be:

Points
A-A Top 9
Fantasyland 8
Scoop 6
23

The expectation of playing it safe would simply be a loss of one point. If we gamble and fail to make it, we will lose six points.

If we determine that on the fourth and final pull that the probabilities of making a flush is 8% and two pair or better is 12%, we can estimate the expected value of gambling: (Note that we have an 80% chance of fouling)

EV Gambling = (80%)(-6) + (8%)(27) + (12%)(23) = -4.8 + 2.16 + 2.76 = .12

So we can see that in this scenario, choosing to take the gamble fares better by around 1.12 points. It is relatively close though, and we should fold if more of our outs are dead, our upside potential is not as great, and/or we don’t have a chance to scoop.

Notice that we did not specify whether or not our opponent was likely to score any royalties or also potentially take a trip to Fantasyland. As discussed in a previous issue, these considerations don’t factor into our decision making as we would be on the hook to pay any bonuses to him regardless if we foul or not. Our opponent’s holding only matters in gauging how many lines he would be likely to win, and also in the determination of how many outs we have.

While it is impossible to be so precise in the heat of the battle, it’s important to be cognizant of the key principles in play. It’s often correct to gamble in POFC, and we should not be afraid to do so when having substantial upside as long as our outs are relatively live. Do your best and review hands afterwards either in a solver or via spreadsheet, where a working knowledge of combinations will often be your best friend. ♠

Kevin Haney is a former actuary but left the corporate job to focus on his passions for poker and fitness. The certified personal trainer and former gym owner resides in 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].