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Let it Ride Appendix
Introduction
Statistics for First Three CardsThe 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 CardsThe 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 CardsThe 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.
SummaryFrom 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.
Go back to Let it Ride
Written by:Michael Shackleford
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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.
Statistics for First Four CardsThe 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 CardsThe 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.
SummaryFrom 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.
Go back to Let it Ride
Written by:Michael Shackleford
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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.
Statistics for Final Five CardsThe 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.
SummaryFrom 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.
Go back to Let it Ride
Written by:Michael Shackleford
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.
SummaryFrom 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.
Go back to Let it Ride
Written by:Michael Shackleford
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.
Go back to Let it Ride
Written by:Michael Shackleford
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