# Flush Fever

## Introduction

Flush Fever is a video poker game I noticed at Figaro's pizza restaurant in Philomath, Oregon, on June 21, 2011. It was on an IGT Game King machine titled "Video Lottery" at the top. Although the game is run by the Oregon Lottery, the cards are dealt as fairly and as randomly as they are in Nevada, per Oregon gaming regulations. In particular, Regulation 177-200-0070.

Later I would find another game in Las Vegas called Flush Fever, but with a different concept. To distinguish the two, I shall call this the Oregon variant and the other the Nevada variant.

## Rules

The game has the following pay table. Due to a \$600 maximum win, the return goes down after six coins. The player may bet up to eight coins but is not rewarded on the royal for the 7th or 8th coin bet.

### Flush Fever Pay Table

Hand Pays
Royal flush 400
Straight flush 100
Four of a kind 25
Full house 10
Flush 7
Straight 5
Three of a kind 2
Two pair 1

In addition, there are the following bonus rules.

1. If the player gets any flush, straight flush, or royal flush, then he shall win 7 Flush Fever Games.
2. Unlike free games, the player must still make a bet equal to the initiating bet for all Flush Fever Games.
3. The suit that triggered the Flush Fever bonus shall be known as the Fever Suit.
4. In additional to any pay table wins, each card in the final hand of a Fever Game in the Fever Suit shall win an amount equal to the initiating bet.
5. Fever Games may earn more Fever Games, up to a total of 56 per bonus.
6. The \$600 maximum win per the Oregon Lottery applies to each individual initiating game and Flush Fever game.

## Analysis

The following table shows the expected return of initial spins. This table does not implicitly count the value of earned bonuses, but the probabilities of each hand reflect the fact that a flush, straight flush, and royal flush are worth about an extra 12 credits due to Fever Spins. The lower right cell shows a return of 63.70% from non-bonus wins.

### Flush Fever Return Table — Initial Spins

Hand Pays Combinations Probability Return
Royal flush 400 399,357,684 0.000020 0.008014
Straight flush 100 3,569,097,180 0.000179 0.017905
Four of a kind 25 38,423,682,000 0.001928 0.048190
Full house 10 195,735,497,760 0.009820 0.098196
Flush 7 498,970,893,696 0.025032 0.175225
Straight 5 286,894,804,944 0.014393 0.071964
Three of a kind 2 1,127,665,371,096 0.056572 0.113144
Two pair 1 2,080,119,126,888 0.104354 0.104354
Nothing 0 15,701,452,685,952 0.787702 0.000000
Total 0 19,933,230,517,200 1.000000 0.636993

The next table shows the expected return of Fever Spins. This table does not implicitly count the value of additional Fever Spins earned in Fever Spins, nor Fever Suit wins. The lower right cell shows a return of 46.84% from non-bonus wins.

### Flush Fever Return Table — Fever Spins

Hand Pays Combinations Probability Return
Royal flush 400 330,791,403 0.000017 0.006638
Straight flush 100 2,763,418,713 0.000139 0.013863
Four of a kind 25 26,515,426,776 0.001330 0.033255
Full house 10 157,550,073,984 0.007904 0.079039
Flush 7 349,728,539,904 0.017545 0.122815
Straight 5 164,126,005,884 0.008234 0.041169
Three of a kind 2 871,582,046,262 0.043725 0.087450
Two pair 1 1,677,077,080,218 0.084135 0.084135
Nothing 0 16,683,557,134,056 0.836972 0.000000
Total 0 19,933,230,517,200 1.000000 0.468364

The average number of Fever Suit cards earned per Fever Spin is 2.075903. Aside from additional Fever Spins earned, the value per Fever Spin is 0.468364 + 2.075903 = 2.544267.

There is a lot that goes into calculating the value of a Flush Fever bonus. Here is my train of thought.

• The Flush Fever bonus starts with 7 free games.
• As the probability table above shows, the probability of a flush (including straight flush and royal flush) within a Flush Fever game is 0.017700.
• The expected final number of free games is 7/(1-7×0.017700) = 7.989970.
• The cap of 56 maximum bonus spins is almost negligible. I show it lowers the expected number of Fever Spins by 0.000639. So the adjusted number of Fever Spins per bonus is 7.989970-0.000639=7.989331.
• The probability of a flush, straight flush, or royal flush in an initial spin is 0.025231.
• The expected number of Fever Spins per initial spin is 0.025231 × 7.989331 = 0.201580.

The next table combines both initial spins and Fever Spins to get an overall return of the game. The lower right cell shows a return of 95.70%.

### Flush Fever Return Table — Combined Return

Type of Spin Spins per Initial Spin Ratio of Spins Win per Spin Expected Win
Base 1.000000 0.832237 0.636993 0.530129
Fever 0.201580 0.167763 2.544267 0.426833
Total 1.201580 1.000000   0.956962

For the advanced reader, who is not satisfied with the 2.075903 average Fever Cards per Fever Spin, the following tables break down the possible outcomes in Fever Spins — poker value crossed by Fever Cards. The next table shows what each outcome in Fever Hands pays, according to the poker value and number of Fever Cards.

### Combinations in Flush Fever Hands

Hand Fever Suit
0 1 2 3 4 5
Royal flush 400 401 402 403 404 405
Straight flush 100 101 102 103 104 105
Four of a kind 25 26 27 28 29 30
Full house 10 11 12 13 14 15
Flush 7 8 9 10 11 12
Straight 5 6 7 8 9 10
Three of a kind 2 3 4 5 6 7
Two pair 1 2 3 4 5 6
Nothing 0 1 2 3 4 5

The next table shows the number of combinations of each hand crossed by number of cards in the Fever Suit for Fever Hands.

### Combinations in Flush Fever HandsExpand

Hand Fever Suit
0 1 2 3 4 5 Total
Royal flush 186,582,030 0 0 0 0 144,209,373 330,791,403
Straight flush 1,478,982,090 0 0 0 0 1,284,436,623 2,763,418,713
Four of a kind 0 15,212,352,972 11,303,073,804 0 0 0 26,515,426,776
Full house 13,396,740,840 72,337,344,390 71,815,988,754 0 0 0 157,550,073,984
Flush 164,758,099,950 0 0 0 0 184,970,439,954 349,728,539,904
Straight 23,789,293,920 44,608,669,992 55,996,972,986 31,117,146,072 8,613,922,914 0 164,126,005,884
Three of a kind 37,919,054,430 333,365,054,208 384,816,777,126 115,481,160,498 0 0 871,582,046,262
Two pair 148,220,109,900 589,856,830,920 682,740,166,932 256,259,972,466 0 0 1,677,077,080,218
Nothing 1,038,682,295,280 3,498,899,059,848 5,839,957,445,298 4,667,194,006,614 1,638,824,327,016 0 16,683,557,134,056
Total 1,428,431,158,440 4,554,279,312,330 7,046,630,424,900 5,070,052,285,650 1,647,438,249,930 186,399,085,950 19,933,230,517,200

The next table shows the probability of each hand crossed by number of cards in the Fever Suit for Fever Hands.

### Probabilities in Flush Fever Hands

Hand Fever Suit
0 1 2 3 4 5 Total
Royal flush 0.000009 0.000000 0.000000 0.000000 0.000000 0.000007 0.000017
Straight flush 0.000074 0.000000 0.000000 0.000000 0.000000 0.000064 0.000139
Four of a kind 0.000000 0.000763 0.000567 0.000000 0.000000 0.000000 0.001330
Full house 0.000672 0.003629 0.003603 0.000000 0.000000 0.000000 0.007904
Flush 0.008265 0.000000 0.000000 0.000000 0.000000 0.009280 0.017545
Straight 0.001193 0.002238 0.002809 0.001561 0.000432 0.000000 0.008234
Three of a kind 0.001902 0.016724 0.019305 0.005793 0.000000 0.000000 0.043725
Two pair 0.007436 0.029592 0.034251 0.012856 0.000000 0.000000 0.084135
Nothing 0.052108 0.175531 0.292976 0.234141 0.082216 0.000000 0.836972
Total 0.071661 0.228477 0.353512 0.254352 0.082648 0.009351 1.000000

The next table shows the return from each hand crossed by number of cards in the Fever Suit for Fever Hands. The lower right cell shows that each Fever Suit hand has a return of 2.54 times the initiating bet.

### Returns in Flush Fever Hands

Hand Fever Suit
0 1 2 3 4 5 Total
Royal flush 0.003744 0.000000 0.000000 0.000000 0.000000 0.002930 0.006674
Straight flush 0.007420 0.000000 0.000000 0.000000 0.000000 0.006766 0.014186
Four of a kind 0.000000 0.019842 0.015310 0.000000 0.000000 0.000000 0.035153
Full house 0.006721 0.039919 0.043234 0.000000 0.000000 0.000000 0.089874
Flush 0.057858 0.000000 0.000000 0.000000 0.000000 0.111354 0.169213
Straight 0.005967 0.013427 0.019665 0.012489 0.003889 0.000000 0.055437
Three of a kind 0.003805 0.050172 0.077221 0.028967 0.000000 0.000000 0.160165
Two pair 0.007436 0.059183 0.102754 0.051424 0.000000 0.000000 0.220797
Nothing 0.000000 0.175531 0.585952 0.702424 0.328863 0.000000 1.792770
Total 0.092951 0.358075 0.844136 0.795303 0.332752 0.121050 2.544267

## Strategy

The Oregon Lottery is kind enough to suggest which cards the player should hold in all video poker games. The cards they suggest to keep are already held. If you accept their advice all the player needs to do is keep pressing the deal/draw button, like a slot player. The player may also override the advice if he wishes.

The help screens say that the advice may not be optimal. However, I trust IGT, the maker of the Game King machine, to get the advice feature right. In Fever Spins, 82% of the return is from Fever Cards. Thus, the player can expect the Fever Spin strategy to be mostly trying to accumulate as many Fever Cards as possible.

## Acknowledgments

• IGT, for helping me with some questions about the rules.
• CrystalMath, for suggesting I base the return on all bets, not just the initial bet.

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