Video Poker Appendix 5
I have always
wondered if I ever did get a royal flush in video poker would it
likely be after holding 3, 4, or some other numbers of cards on the
deal. This section answers that question, not just for royal flushes
but every hand. Everything is based on full
pay deuces wild and optimal strategy. When the player holds four
deuces the program holds all five cards.
| Probability of Each Hand by Number
of Cards Held |
| Hand |
5 held |
4 held |
3 held |
2 held |
1 held |
0 held |
Total |
| Natural royal |
0.069692 |
0.318806 |
0.544452 |
0.055006 |
0 |
0.012044 |
1 |
| 4 deuces |
0.090666 |
0 |
0.340158 |
0.373651 |
0.169288 |
0.026238 |
1 |
| Wild royal |
0.102843 |
0.367608 |
0.294709 |
0.120801 |
0.090096 |
0.023943 |
1 |
| 5 of a kind |
0.054802 |
0.286223 |
0.357392 |
0.211498 |
0.072459 |
0.017626 |
1 |
| Straight flush |
0.146441 |
0.291944 |
0.204802 |
0.15898 |
0.153092 |
0.044742 |
1 |
| 4 of a kind |
0 |
0.160684 |
0.367912 |
0.297955 |
0.13186 |
0.041588 |
1 |
| Full house |
0.229675 |
0 |
0.307975 |
0.357368 |
0.060472 |
0.04451 |
1 |
| Flush |
0.207739 |
0.338909 |
0.176969 |
0.075126 |
0.132157 |
0.0691 |
1 |
| Straight |
0.273453 |
0.258743 |
0.093541 |
0.095039 |
0.189034 |
0.090191 |
1 |
| 3 of a kind |
0 |
0.011344 |
0.332021 |
0.374545 |
0.183876 |
0.098214 |
1 |
| Nothing |
0 |
0.119155 |
0.068301 |
0.417122 |
0.116461 |
0.27896 |
1 |
The above table shows that 54% of royals will come after holding
3 to a royal, and 32% after holding 4 to a royal. Four deuces will
come along 34% of the time holding 3 deuces and 37% holding 2
deuces.
The next table shows the overall probability of each hand
according to the number of cards held on the deal
| Probability of Each Hand and Number
of Cards Held |
| Hand |
5 held |
4 held |
3 held |
2 held |
1 held |
0 held |
Total |
| Natural royal |
0.000002 |
0.000007 |
0.000012 |
0.000001 |
0 |
0 |
0.000022 |
| 4 deuces |
0.000018 |
0 |
0.000069 |
0.000076 |
0.000034 |
0.000005 |
0.000204 |
| Wild royal |
0.000185 |
0.00066 |
0.000529 |
0.000217 |
0.000162 |
0.000043 |
0.001796 |
| 5 of a kind |
0.000175 |
0.000916 |
0.001144 |
0.000677 |
0.000232 |
0.000056 |
0.003202 |
| Straight flush |
0.000603 |
0.001203 |
0.000844 |
0.000655 |
0.000631 |
0.000184 |
0.00412 |
| 4 of a kind |
0 |
0.010435 |
0.023892 |
0.019349 |
0.008563 |
0.002701 |
0.064938 |
| Full house |
0.004876 |
0 |
0.006538 |
0.007587 |
0.001284 |
0.000945 |
0.021229 |
| Flush |
0.003444 |
0.005619 |
0.002934 |
0.001246 |
0.002191 |
0.001146 |
0.016581 |
| Straight |
0.015468 |
0.014636 |
0.005291 |
0.005376 |
0.010693 |
0.005102 |
0.056564 |
| 3 of a kind |
0 |
0.003228 |
0.094475 |
0.106575 |
0.052321 |
0.027946 |
0.284544 |
| Nothing |
0 |
0.065154 |
0.037347 |
0.228082 |
0.063681 |
0.152535 |
0.5468 |
| Total |
0.024771 |
0.101858 |
0.173075 |
0.36984 |
0.139791 |
0.190664 |
1 |
The final table summarizes this information, showing the
probability of holding each number of cards, the expected value, and
the contribution to the total return. This table shows among other
things that the most common play is holding 2 cards and the most of
the return comes when holding 3 cards.
| Probabilities and Return by Number
of Cards Held |
| Cards Held |
Probability |
Exp. Value |
Tot. Return |
| 5 |
0.024771 |
2.828083 |
0.070056 |
| 4 |
0.101858 |
1.400165 |
0.142618 |
| 3 |
0.173075 |
1.799580 |
0.311463 |
| 2 |
0.369840 |
0.748947 |
0.276991 |
| 1 |
0.139791 |
1.036199 |
0.144852 |
| 0 |
0.190664 |
0.323298 |
0.061641 |
| Total |
1 |
|
1.007620 |
