Let it Ride, Appendix 1

Last update: Nov. 21, 2009

Statistics for First Three Cards

The following table shows all possible combinations of three cards in which it is advantageous to raise. First, here is additional explanation of the contents of each column.

Hand: Hand formed with the first 3 cards
Number: Number of combinations of the corresponding hand out of 20492 possible.
Probability: Probability of forming this 3 card hand
Total return: This is the total return expected over all possible 1176 combinations of 2 additional cards.
Average return: This is the average return of the given hand, which equals column 4 divided by 1176
Expected return: This is the weighted return of the given hand, which equals the product of column 3 and column 5.

Let it Ride - Probability and Returns of First Three Cards
Hand	Number	Probability	Total return	Average return	Expected return
High pair	1440	0.06515837	1689	1.43622449	0.09358205
Three of a kind	52	0.00235294	6360	5.40816327	0.01272509
3 to a straight flush, spread 3, 0 high cards	20	0.00090498	252	0.21428571	0.00019392
3 to a straight flush, spread 3, 1 high card	4	0.00018100	480	0.40816327	0.00007388
3 to a straight flush, spread 3, 2 high cards	4	0.00018100	708	0.60204082	0.00010897
3 to a straight flush, spread 4, 1 high card	12	0.00054299	198	0.16836735	0.00009142
3 to a straight flush, spread 4, 2 high cards	12	0.00054299	426	0.3622449	0.00019669
3 to a straight flush, spread 5, 2 high cards	24	0.00108597	144	0.12244898	0.00013298
3 to a royal flush, queen highest	4	0.00018100	1736	1.47619048	0.00026718
3 to a royal flush, king highest	12	0.00054299	1454	1.23639456	0.00067135
3 to a royal flush, ace highest	24	0.00108597	1172	0.99659864	0.00108228
Everything else	20492	0.927239819	0	0	0
Total	22100	1			0.10912580

From the table above it can be seen that the player has a probability of 0.07276018 of raising and has an expected return of 0.10912580 units.

Statistics for First Four Cards

The following table shows all possible combinations of four cards in which it is advantageous to raise.

Let it Ride - Probability and Returns of First Four Cards
Hand	Number	Probability	Total return	Average return	Expected return
High pair	31680	0.11701912	58	1.20833333	0.1413981
Two pair	2808	0.01037215	132	2.75000000	0.02852341
Three of a kind	2496	0.00921969	215	4.47916667	0.04129652
Four of a kind	13	0.00004802	2400	50.00000000	0.00240096
4 to a flush, 0 high cards	212	0.00078308	33	0.68750000	0.00053837
4 to a flush, 1 high card	1084	0.00400406	39	0.81250000	0.0032533
4 to a flush, 2 high cards	1100	0.00406316	45	0.93750000	0.00380922
4 to a flush, 3 high cards	300	0.00110814	51	1.06250000	0.00117739
4 to an outside straight, 0 high cards	1260	0.00465417	0	0	0
4 to an outside straight, 1 high card	252	0.00093083	6	0.12500000	0.00011635
4 to an outside straight, 2 high cards	252	0.00093083	12	0.25000000	0.00023271
4 to an outside straight, 3 high cards	252	0.00093083	18	0.37500000	0.00034906
4 to an outside straight, 4 high cards	252	0.00093083	24	0.50000000	0.00046542
4 to an inside straight, 4 high cards	1008	0.00372334	0	0	0
4 to an outside straight flush, 0 high cards	20	0.00007388	453	9.43750000	0.0006972
4 to an outside straight flush, 1 high card	4	0.00001478	459	9.56250000	0.00014129
4 to an outside straight flush, 2 high cards	4	0.00001478	465	9.68750000	0.00014313
4 to an outside straight flush, 3 high cards	4	0.00001478	471	9.81250000	0.00014498
4 to an inside straight flush, 0 high cards	48	0.0001773	243	5.06250000	0.00089759
4 to an inside straight flush, 1 high card	32	0.0001182	249	5.18750000	0.00061317
4 to an inside straight flush, 2 high cards	16	0.0000591	255	5.31250000	0.00031397
4 to an inside straight flush, 3 high cards	16	0.0000591	261	5.43750000	0.00032136
4 to an outside royal flush	4	0.00001478	1277	26.60416667	0.00039308
4 to an inside royal flush	16	0.0000591	1067	22.22916667	0.00131376
Everything else	227592	0.84067596	0	0	0
Total	270725	0.15932404			0.22854034

From the table above it can be seen that the player has a probability of 0.15932404 of raising (including on even money bets) and has an expected return of 0.22854034 units.

Statistics for Final Five Cards

The following table shows all possible combinations of five cards.

Let it Ride - Probability and Return of All Five Cards
Hand	Number	Probability	Total return	Average return
Royal flush	4	0.00000154	1000	0.00153908
Straight flush	36	0.00001385	200	0.00277034
Four of a kind	624	0.00024010	50	0.01200480
Full house	3744	0.00144058	11	0.01584634
Flush	5108	0.00196540	8	0.01572321
Straight	10200	0.00392465	5	0.01962323
Three of a kind	54912	0.02112845	3	0.06338535
Two pair	123552	0.04753902	2	0.09507803
High pair	422400	0.16252655	1	0.16252655
Nothing	1978380	0.76121987	-1	-0.76121987
Total	2598960	0.238780127		-0.37272294

There is a mandatory 1 unit bet on the final five cards. The expected return based on the final five cards is -0.37272294 units.

Summary

From these tables is can be seen the average total amount bet is 0.07276018 + 0.15932404 + 1 = 1.23208422. The average net gain is 0.10912580 + 0.22854034 - 0.372722935 = -0.035056792.

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