The Mathematics of 'Jacks or Better'

Jacks or Better (JoB), specifically the 9/6 Full Pay variant, serves as the mathematical foundation of video poker. Unlike slot machines, JoB operates on known probabilities derived from a fair 52-card deck, making it one of the most transparent games in any casino.

The Numbers Behind the Game

When you're dealt a hand of video poker, you're working with precise mathematics:

2,598,960 possible initial 5-card hands (52 choose 5)

32 possible hold combinations per hand (2^5)

99.54% Return to Player (RTP) with optimal strategy and maximum coins

This transparency is what distinguishes video poker from slot machines. Every probability can be calculated, and every decision has a mathematically optimal answer.

Understanding the 9/6 Paytable

The "9/6" designation refers to the payouts for Full House (9 coins) and Flush (6 coins) per coin wagered:

HandPayout (per coin)ProbabilityRoyal Flush800 (250 base × 5th coin bonus)0.00247%Straight Flush500.01093%Four of a Kind250.2363%Full House91.1512%Flush61.1015%Straight41.1229%Three of a Kind37.4449%Two Pair212.9279%Jacks or Better121.4585%

The Critical Fifth Coin

One of the most important mathematical features of video poker is the Royal Flush bonus on the 5th coin. Playing four coins, a Royal Flush pays 250× each coin (1,000 total). Playing five coins, it pays 800× the fifth coin (4,000 total).

This disproportionate bonus accounts for over 1% of the total RTP. Players who don't bet maximum coins are effectively playing a 98.4% game instead of a 99.5% game—a significant difference over time.

Expected Value Maximization

Every decision in video poker has an Expected Value (EV)—the average return you can expect from each possible hold choice. Optimal strategy is simply choosing the hold pattern with the highest EV.

For example, if you're dealt:

K♠ K♦ Q♥ J♣ 10♠

Your options include:

Hold the pair of Kings (EV: ~1.54)

Hold the 4-card straight (K-Q-J-10) (EV: ~0.87)

Hold the 4-card royal draw (K-Q-J-10 suited) (EV: ~1.29 if suited)

The mathematically correct play is to hold the pair—not because it's guaranteed to win, but because over millions of hands, this choice maximizes your return.

The Penalty Card Phenomenon

Penalty cards are discards that alter the probability of completing straights or flushes. Consider holding a 3-card royal (e.g., Q♥ J♥ 10♥) when you have a 4-card straight (e.g., Q♥ J♥ 10♥ 9♠).

If you discard the 9♠, you're also discarding a card that could complete a straight. This "penalty" slightly reduces the EV of the royal draw. These subtle interactions are why intuition often fails against mathematical rigor.

Short Pay Degradation

Casinos adjust the house edge by modifying payout tables. Common "short pay" variants include:

VersionFull House/FlushRTP9/6 (Full Pay)9/699.54%8/58/597.30%7/57/596.15%6/56/595.00%

The difference between 9/6 and 8/5 may seem small (one coin on two hands), but it represents a 2.24% swing in house edge—the difference between a skilled game and a pure gambling exercise.

Variance Considerations

9/6 Jacks or Better is considered a low variance game. The frequent payouts (Jacks or Better hits ~21% of the time) mean your bankroll fluctuates less dramatically than in high-variance games like Double Double Bonus. This makes it ideal for players seeking extended play sessions with moderate risk.