# 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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