Having trouble with 4 Card Poker's math

I am trying to figure out the probability charts for Four Card Poker to verify Shufflemaster's math on the Bad Beat wager.

The first thing I am hitting a wall on is the Player's hands probabilities. I can figure out why there are 624 ways to make a hand with Quads using a 5 card hand. There are 13 possible ways to make quads, and each one could use any of the remaining 48 cards to reach 624.

The next hand (a Straight Flush) is where I already start to bog down. There are 11 ways to make a 4 card SF for each suit, so 11 * 4 = 44. There again should be 48 remaining cards that will finish out the 5 card hand, so 44*48= 2112. But on the charts for the game, it states that there are 2072 ways to make the 4 Card Straight Flushes. If the 5th card ends up creating a 5 card SF, that doesn't matter on this game since only 4 cards play. Why is there a 40 card difference.

I start having similar discrepancies when I try to figure out the other hands when compared to the charts from Shufflemaster (and WOO). I can't begin to work on the Bad Beat wager math until I figure out why there is this discrepancy. Please help.

Here is the breakdown for the possible hand combinations (for the Player that receives 5 cards to play with) from Shufflemaster:
4 of a Kind = 624
Straight Flush = 2072
Three of a Kind = 58656
Flush = 114616
Straight = 101808
2 Pair = 123552
Other = 2197632
Total = 2598960

I'm going to have to do the same for the Dealer's 6 card hand, too if anyone feels ambitious enough to help.

Thanks

I'm going to try to post the chart for the charts from Shufflemaster above, if it isn't there, then I didn't figure out how to post an image :)

Cant figure it out

The reason for the disparity with Straight Flushes is because of hands that have a 5-card straight flush, such as 5s 6s 7s 8s 9s: you are counting those as having 2 straight flushes (5678 and 6789), but only the higher one counts.

11 * 4 = 44 * 48 discards = 2112

however,

there are 4 suits * 10 rank patterns = 40 five-card straight flushes to subtract out to prevent double-counting

2112 - 40 = 2072

Expect the same situation with straights, and be careful with flushes too.

That makes sense. That didn't even cross my mind. Thanks.

Since we're on the subject, do you agree with their methodology for getting the odds for their Bad Beat wager?

From Shufflemaster "The Player hand and the Dealer hand are mostly independent events. While there is some impact
by the rank of one hand on the other, this impact is relatively minimal. In order to get a very
accurate approximation of the payback, a distribution of each the Player’s hand and the Dealer’s
hand was created using a computer program. Each hand has a unique distribution because the
Player is dealt 5 cards and the Dealer is dealt 6 cards. Based on these distributions, the
probability of any rank of hand being a ‘bad beat’ can be calculated by multiplying by the
probability of that rank of hand times the probability of the other hand having a higher rank
added to ½ the identical rank of the other hand.

For instance, the probability that the Player will have a Straight that is beat is as follows:
Probability of the Player having a Straight times (Probability of the Dealer having a Flush or Better
plus ½ times the probability of the Dealer having a Straight).
We use ½ the probability of the Dealer having a Straight because on average ½ of the Straights
the Player has will be beaten by a Dealer Straight. We perform a similar calculation for each
‘winning’ hand (for both Dealer and Player) and multiply each by the payout for that hand. The
total represents the payback for the sidebet"

It is somewhat crude, but much faster than using brute force, and shouldn't be too far off from the results of a brute-force analysis. So, in short, yes. However, since it was relatively quick to do, I just ran a brute-force analysis. I assumed that a player-dealer tie (which is a player win for the main bet) counted as a beat for this bet. Here are the results:

Lowest Hand	Payoff	Combinations	Probability	Hit Frequency	Return
Four of a Kind	25,000	6,198,192	0.000000	4,502,365.00	0.005553
Straight Flush	10,000	92,919,624	0.000003	300,329.70	0.033297
Three of a Kind	100	33,828,086,464	0.001212	824.95	0.121219
Flush	25	274,640,888,696	0.009841	101.61	0.246036
Straight	15	426,287,587,656	0.015276	65.46	0.229133
Two Pair	4	826,240,735,944	0.029607	33.78	0.118430
One Pair	-1	10,305,886,028,016	0.369300	2.71	-0.369300
High Card	-1	16,039,540,279,488	0.574760	1.74	-0.574760
Totals		27,906,522,724,080	1.000000	17.88	-0.190392

The house edge is 19.04%.

The interesting thing about your calculations is that your house edge of 19.04% is different than the advertised 20.3% that Shufflemaster says it is. Do you mind if I share your findings with them to find out what their response is?

Quote: brianparkes

The interesting thing about your calculations is that your house edge of 19.04% is different than the advertised 20.3% that Shufflemaster says it is. Do you mind if I share your findings with them to find out what their response is?

There are two possible explanations for the difference:

(a) because my analysis was exact while theirs was crude

and/or

(b) because their analysis treated tied hands as a loss for this bet, which I did not

I recalculated the analysis treating ties as a loss, but the house edge under that methodology is 19.47% as shown below:

Lowest Hand	Payoff	Combinations	Probability	Hit Frequency	Return
Four of a Kind	25,000	6,198,192	0.000000	4,502,365.00	0.005553
Straight Flush	10,000	87,995,040	0.000003	317,137.45	0.031532
Three of a Kind	100	33,828,086,464	0.001212	824.95	0.121219
Flush	25	274,446,442,424	0.009834	101.68	0.245862
Straight	15	422,159,689,608	0.015128	66.10	0.226915
Two Pair	4	826,142,341,032	0.029604	33.78	0.118416
One Pair	-1	10,304,085,252,576	0.369236	2.71	-0.369236
High Card	-1	16,033,053,643,680	0.574527	1.74	-0.574527
Tie	-1	12,713,075,064	0.000456	2,195.10	-0.000456
Totals		27,906,522,724,080	1.000000	17.93	-0.194722

So, their analysis method (which is like dealing the player hand from a separate deck as the dealer's hand) must be the reason for the difference. With such a high house advantage either way, I doubt anyone cares.

And of course, these figures all assume that my analysis was done correctly, which it hopefully was.

I emailed Shufflemaster about what happens in the case of a Tie. They stated that ties lose.
"Ties lose, the essence of a Bad Beat wager is to pay a hand that takes a bad beat, matching hands tie so there is no winner. This basically does not have a beaten hand in this case unless the case is a higher two pair, straight, flush, etc. the easiest way to see it is “the beat gets paid” – the hand beaten is the hand that is paid according to the pay table".
Thanks for all your work JB. I can now let everyone know that we just had a string of variance that resulted in such a (perceived) high frequency of Trips over Trips (or better) payouts and it should not be considered something that will happen that often on a regular basis.

JB,

I ran a full combinatorial analysis. It took 4.5 hours on my three-year old Ubuntu box, running 12 concurrent processes. I have to learn how to use the "cloud" or get a faster computer.

I get exactly the same results:



I personally think it's a big deal that Shuffle Master has published incorrect numbers for both the house edge of the main game and for this side bet. This faulty math appears to date back to 2003. See the Four Card Poker Training Manual, first published in 2003 but revised (and not fixed) on 5/12/2011. This document gives the house edge on the main game as 1.58%. I get about 2.9%.