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Let it Ride Appendix
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 |
Average |
Expected |
|
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.
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 |
Average |
Expected |
|
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.
The following table shows all possible combinations of five cards.
Let it Ride - Probability and Return of All Five Cards
|
Hand |
Number |
Probability |
Total |
Average |
|
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.
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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