# Video Poker — Probability of winning hand by number of cards held

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

Finally, here are the number of royal flush combinations in 9/6 Jacks or Better, out of a possible 19,933,230,517,200, by the number of cards held. The right column shows the ratio of all royals according to number of cards held.

### 9/6 Jacks Royal Combinations

Cards Held Combinations Probability
0 1,426,800 0.002891
1 16,805,604 0.034053
2 96,804,180 0.196154
3 195,055,740 0.395240
4 152,741,160 0.309498
5 30,678,780 0.062164
Total 493,512,264 1.000000

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