# Multi-Strike Poker Appendix 3

## Introduction

The purpose of this page is to show my detailed calculations for the Full Pay Deuces Wild (FPDW) pay table in Multi-Strike Poker.

The following table shows the probability of getting a free ride at each level.

### FPDW Free Ride Probabilities

Level Probability
3 7.80%
2 7.80%
1 8.25%

The following table shows the return table for playing level 4 of FPDW. This also reflects the probabilities for conventional FPDW, and for any level where a Free Ride card was awarded. This wins are 16 times those of level 1. The lower right cell shows the value of attaining level 4 is 8.060957 times the bet per line.

### FPDW — Level 4 Return Table

Event Pays Combinations Probability Return
Natural royal flush 6400 440,202,756 0.000022 0.141337
4 deuces 1600 4,060,462,824 0.000204 0.325925
Wild royal flush 200 35,796,957,696 0.001796 0.359169
5 of a kind 120 63,818,309,856 0.003202 0.384192
Straight flush 72 82,122,481,680 0.004120 0.296631
4 of a kind 40 1,294,427,430,576 0.064938 2.597527
Full house 24 423,165,297,240 0.021229 0.509499
Flush 16 330,506,232,804 0.016581 0.265291
Straight 16 1,127,512,239,276 0.056564 0.905031
3 of a kind 8 5,671,888,316,712 0.284544 2.276355
Nothing 0 10,899,492,585,780 0.546800 0.000000
Total   19,933,230,517,200 1.000000 8.060957

The following table shows the return table for playing level 3 of FPDW, when a Free Ride was not awarded. The wins are 4 times those of level 1, plus the expected value of 8.060957 of level 4 for any paying hand. The lower right cell shows the value of attaining level 3 with no Free Ride is 15.044824 times the bet per line.

### FPDW — Level 3 Return Table — No Free Ride

Event Pays Combinations Probability Return
Natural royal flush 3208.060957 417,777,504 0.000021 0.067237
4 deuces 808.0609569 4,110,107,700 0.000206 0.166617
Wild royal flush 108.0609569 32,932,616,280 0.001652 0.178533
5 of a kind 68.0609569 64,358,413,080 0.003229 0.219748
Straight flush 44.0609569 76,886,019,288 0.003857 0.169951
4 of a kind 28.0609569 1,311,734,068,716 0.065806 1.846590
Full house 20.0609569 427,803,152,904 0.021462 0.430544
Flush 16.0609569 315,562,064,400 0.015831 0.254260
Straight 16.0609569 1,048,879,491,840 0.052620 0.845122
3 of a kind 12.0609569 5,825,829,039,480 0.292267 3.525022
Nothing 0 10,824,717,766,008 0.543049 0.000000
Total   19,933,230,517,200 1.000000 7.703625

The following table shows the return table for playing level 4 of FPDW, when a Free Ride was awarded. The wins are 8 times those of level 1, plus the expected value of 8.060957 of level 4 for any hand. The lower right cell shows the value of attaining level 3 with a Free Ride is 23.523527 times the bet per line.

### FPDW — Level 3 Return Table — Free Ride

Event Pays Combinations Probability Return
Natural royal flush 3208.060957 440,202,756 0.000022 0.070846
4 deuces 808.060957 4,060,462,824 0.000204 0.164605
Wild royal flush 108.060957 35,796,957,696 0.001796 0.194061
5 of a kind 68.060957 63,818,309,856 0.003202 0.217904
Straight flush 44.060957 82,122,481,680 0.004120 0.181526
4 of a kind 28.060957 1,294,427,430,576 0.064938 1.822227
Full house 20.060957 423,165,297,240 0.021229 0.425877
Flush 16.060957 330,506,232,804 0.016581 0.266301
Straight 16.060957 1,127,512,239,276 0.056564 0.908479
3 of a kind 12.060957 5,671,888,316,712 0.284544 3.431877
Nothing 8.060957 10,899,492,585,780 0.546800 4.407732
Total   19,933,230,517,200 1.000000 12.091435

The next table is a weighted average of the two level 3 tables, according to the Free Ride probability. The lower right cell shows the value of attaining level 3 is 8.045874 times the bet per level, before it is determined if a Free Ride is awarded.

### FPDW — Level 3 Combined Return Table

Free Ride Probability Expected Return Expected Value
Yes 0.078 12.091435 0.943132
No 0.922 7.703625 7.102742
Total 1 - 8.045874

The following table shows the return table for playing level 2 of FPDW, when a Free Ride was not awarded. The wins are 2 times those of level 1, plus the expected value of 8.045874 of level 3 for any paying hand. The lower right cell shows the value of attaining level 2 with no Free Ride is 5.688897 times the bet per line.

### FPDW — Level 2 Return Table — No Free Ride

Event Pays Combinations Probability Return
Natural royal flush 1608.045874 352,599,540 0.000018 0.028445
4 deuces 408.0458742 4,135,053,204 0.000207 0.084647
Wild royal flush 58.04587425 28,881,001,728 0.001449 0.084102
5 of a kind 38.04587425 64,679,487,096 0.003245 0.123452
Straight flush 26.04587425 75,959,998,392 0.003811 0.099254
4 of a kind 18.04587425 1,319,770,944,444 0.066210 1.194810
Full house 14.04587425 429,993,794,952 0.021572 0.302993
Flush 12.04587425 308,214,731,724 0.015462 0.186258
Straight 12.04587425 1,049,449,618,236 0.052648 0.634194
3 of a kind 10.04587425 5,854,925,226,024 0.293727 2.950743
Nothing 0 10,796,868,061,860 0.541652 0.000000
Total   19,933,230,517,200 1.000000 5.688897

The following table shows the return table for playing level 2 of FPDW, when a Free Ride was awarded. The wins are 2 times those of level 1, plus the expected value of 8.045874 of level 3 for any hand. The lower right cell shows the value of attaining level 2 with a Free Ride is 10.061113 times the bet per line.

### FPDW — Level 3 Return Table — Free Ride

Event Pays Combinations Probability Return
Natural royal flush 1608.045874 440,202,756 0.000022 0.035512
4 deuces 408.045874 4,060,462,824 0.000204 0.083120
Wild royal flush 58.045874 35,796,957,696 0.001796 0.104241
5 of a kind 38.045874 63,818,309,856 0.003202 0.121808
Straight flush 26.045874 82,122,481,680 0.004120 0.107306
4 of a kind 18.045874 1,294,427,430,576 0.064938 1.171866
Full house 14.045874 423,165,297,240 0.021229 0.298182
Flush 12.045874 330,506,232,804 0.016581 0.199729
Straight 12.045874 1,127,512,239,276 0.056564 0.681368
3 of a kind 10.045874 5,671,888,316,712 0.284544 2.858497
Nothing 8.045874 10,899,492,585,780 0.546800 4.399485
Total   19,933,230,517,200 1.000000 10.061113

The next table is a weighted average of the two level 2 tables, according to the Free Ride probability. The lower right cell shows the value of attaining level 2 is 6.029930 times the bet per level, before it is determined if a Free Ride is awarded.

### FPDW — Level 2 Combined Return Table

Free Ride Probability Expected Return Expected Value
Yes 0.078 10.061113 0.784767
No 0.922 5.688897 5.245163
Total 1 - 6.029930

The following table shows the return table for playing level 1 of FPDW, when a Free Ride was not awarded. The wins include the expected value of 6.029930 of level 2 for any paying hand. The lower right cell shows the value of attaining level 1 with no Free Ride is 3.765257 times the bet per line.

### FPDW — Level 1 Return Table — No Free Ride

Event Pays Combinations Probability Return
Natural royal flush 806.02993 311,564,400 0.000016 0.012599
4 deuces 206.02993 4,137,281,292 0.000208 0.042763
Wild royal flush 31.02993004 27,031,420,440 0.001356 0.042080
5 of a kind 21.02993004 64,785,708,888 0.003250 0.068350
Straight flush 15.02993004 74,427,700,308 0.003734 0.056120
4 of a kind 11.02993004 1,322,859,685,584 0.066365 0.731996
Full house 9.02993004 431,437,958,172 0.021644 0.195445
Flush 8.02993004 304,697,744,388 0.015286 0.122745
Straight 8.02993004 1,048,553,327,628 0.052603 0.422401
3 of a kind 7.02993004 5,871,597,609,348 0.294563 2.070759
Nothing 0 10,783,390,516,752 0.540976 0.000000
Total   19,933,230,517,200 1.000000 3.765257

The following table shows the return table for playing level 1 of FPDW, when a Free Ride was awarded. The wins include the expected value of 6.029930 of level 2 for any hand. The lower right cell shows the value of attaining level 2 with a Free Ride is 7.037550 times the bet per line.

### FPDW — Level 1 Return Table — Free Ride

Event Pays Combinations Probability Return
Natural royal flush 806.029930 440,202,756 0.000022 0.017800
4 deuces 206.029930 4,060,462,824 0.000204 0.041969
Wild royal flush 31.029930 35,796,957,696 0.001796 0.055725
5 of a kind 21.029930 63,818,309,856 0.003202 0.067330
Straight flush 15.029930 82,122,481,680 0.004120 0.061921
4 of a kind 11.029930 1,294,427,430,576 0.064938 0.716263
Full house 9.029930 423,165,297,240 0.021229 0.191698
Flush 8.029930 330,506,232,804 0.016581 0.133142
Straight 8.029930 1,127,512,239,276 0.056564 0.454209
3 of a kind 7.029930 5,671,888,316,712 0.284544 2.000327
Nothing 6.029930 10,899,492,585,780 0.546800 3.297166
Total   19,933,230,517,200 1.000000 7.037550

The next table is a weighted average of the two level 1 tables, according to the Free Ride probability. The lower right cell shows the value of attaining level 1 is 4.035221 times the bet per level, before it is determined if a Free Ride is awarded.

### FPDW — Level 1 Combined Return Table

Free Ride Probability Expected Return Expected Value
Yes 0.0825 7.037550 0.580598
No 0.9175 3.765257 3.454623
Total 1 - 4.035221

The player must bet four lines, so to get the expected return of the game divide the total return of 4.035221 by 4 to get 1.008805. In single-level mode the return for FPDW is 1.007620. So, betting all five levels increases the return by 0.12%.