# Video Poker with Replacement

## Introduction

In November, 2021, a reader sent me evidence of a video poker game where he was sometimes getting back the same card on the draw that he discarded after the deal. I can't call this cheating, because the rule screen did say, "When you click 'deal,' cards with 'held' will remain in play with the others shuffled back into the deck." The game was developed by Pure RNG Gaming. This page analyzes the odds of said game.

## Rules

The following are the rules, as found as Internet casinos using this game.

1. A single 52-card deck is used.
2. After making a bet, the player is dealt five cards from the deck.
3. The player may hold any cards he wishes, the rest to be discarded.
4. The discards are placed back with the remaining 47 undealt cards.
5. Replacement cards are dealt from the deck.
6. The player is paid based on the poker value of his final hand and the pay table below.

The following table shows what each hand pays. Wins are on a "for one" basis.

### Pay Table

Poker Hand Pays
Royal flush 800
Straight flush 50
Four of a kind 25
Full house 9
Flush 6
Straight 4
Three of a kind 3
Two pair 2
Jacks or better 1
Nothing 0

## Analysis

The following table shows the probability and contribution to the return for all possible events, assuming optimal player strategy for this game. The lower right cell shows an expected return of 96.60%.

### Return Table — Optimal Strategy

Poker Hand Pays Combinations Probability Return
Royal flush 800 734,636,352 0.000022 0.017402
Straight flush 50 3,514,720,736 0.000104 0.005203
Four of a kind 25 73,614,865,896 0.002180 0.054492
Full house 9 378,547,499,304 0.011209 0.100877
Flush 6 366,123,839,872 0.010841 0.065044
Straight 4 371,569,671,840 0.011002 0.044008
Three of a kind 3 2,402,049,989,664 0.071123 0.213370
Two pair 2 4,354,485,731,856 0.128934 0.257868
Jacks or better 1 7,014,355,671,540 0.207691 0.207691
Nothing 0 18,807,968,780,940 0.556894 0.000000
Total   33,772,965,408,000 1.000000 0.965957

The following table shows the probability and contribution to the return for all possible events, assuming the player played optimal strategy for conventional 9/6 jacks or poker, without replacement. The lower right cell shows an expected return of 96.59%.

### Return Table — Without Replacement Strategy

Poker Hand Pays Combinations Probability Return
Royal flush 800 767,547,516 0.000023 0.018181
Straight flush 50 3,524,021,144 0.000104 0.005217
Four of a kind 25 73,598,753,592 0.002179 0.054481
Full house 9 378,491,782,968 0.011207 0.100863
Flush 6 362,727,142,360 0.010740 0.064441
Straight 4 371,824,584,732 0.011010 0.044038
Three of a kind 3 2,401,771,851,648 0.071115 0.213346
Two pair 2 4,354,714,729,872 0.128941 0.257882
Jacks or better 1 7,007,119,858,020 0.207477 0.207477
Nothing 0 18,818,425,136,148 0.557204 0.000000
Total Total 33,772,965,408,000 1.000000 0.965925

## Comparison to Video Poker without Replacement

The following table shows the probability and contribution to the return for conventional jacks or better video poker with the same pay table where discards are not replaced in the deck. The lower right cell shows an expected return of 99.54%.

### Return Table — No Replacement

Poker Hand Pays Combinations Probability Return
Royal flush 800 493,512,264 0.000025 0.019807
Straight flush 50 2,178,883,296 0.000109 0.005465
Four of a kind 25 47,093,167,764 0.002363 0.059064
Full house 9 229,475,482,596 0.011512 0.103610
Flush 6 219,554,786,160 0.011015 0.066087
Straight 4 223,837,565,784 0.011229 0.044917
Three of a kind 3 1,484,003,070,324 0.074449 0.223346
Two pair 2 2,576,946,164,148 0.129279 0.258558
Jacks or Better 1 4,277,372,890,968 0.214585 0.214585
Nothing 0 10,872,274,993,896 0.545435 0.000000
Total 0 19,933,230,517,200 1.000000 0.995439

The cost of replacing the discards back in the deck, assuming optimal strategy for each game, is 2.948%. If the player did not use optimal strategy in the replacement game, but used the proper strategy without replacement, there would be an additional cost of errors of 0.003%, bringing the total cost to 2.951%.

## Strategy

As explained above, conventional 9/6 Jacks or Better strategy can be used at a cost of error of only 0.003%.

## Ethical Issues

Video poker came to casinos in 1989, known then as "draw poker." For 32 years now, the game has been dealt the same way, discarding the discards and dealing replacement cards from the 47 cards left in the deck. Players are used to this rule. Few players need to review the rules page of the game, because the rules are so widely understood.

As shown above, the rule to put the discards back in the deck costs the player 2.95%. If it were not for the fact that this rule is disclosed, I would say it is cheating. However, the rule is disclosed in the rules screen. All I can do is give my readers a firm warning about it. If any player felt such play were unethical, I would have no compunction to not patronize PureRNG ( the software company that made it) nor any casino that offers the game.