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Last Updated: November 21, 2009

Let it Ride Appendix


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 overall 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 highcards

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 highcards

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 highcards

12

0.00054299

426

0.3622449

0.00019669

3 to a straight flush, spread 5, 2 highcards

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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