Royal flush odds — the exact math
The drought table
Probability of hitting zero royals in N optimally-played hands:
| Hands | ≈ hours (600/hr) | P(no royal) |
|---|---|---|
| 10,000 | 17 | 78.1% |
| 20,000 | 33 | 60.9% |
| 40,391 | 67 | 36.8% |
| 80,781 | 135 | 13.5% |
| 121,172 | 202 | 5.0% |
| 200,000 | 333 | 0.7% |
The 36.8% at one full cycle is the classic e⁻¹ — droughts aren't bad luck, they're the geometry of rare events. Bankroll for the variance, not the average.
Why chasing changes the odds
Royal frequency is a function of strategy. Hold 3-to-a-royal over a high pair everywhere and you'll see more royals — and lose more money, because the EV you burn between them exceeds what the extra royals pay. The trainer keeps you on the line where the 99.5439% lives.
FAQ
What are the odds of hitting a royal flush in video poker?
Playing 9/6 Jacks or Better with optimal strategy, a royal flush arrives once every 40390.5 hands on average — about 67 hours at 600 hands/hour. Being DEALT a pat royal is far rarer: 1 in 649,740.
Can I go 100,000 hands without a royal?
Yes, and it isn't even unlikely: the probability of zero royals in 100,000 optimally-played hands is about 8.4%. Droughts twice the average cycle happen to every regular player.
Do the odds change if I play badly?
Yes — strategy changes royal frequency. Chasing royals too hard raises royal frequency but burns EV elsewhere; ignoring royal draws lowers it. Our numbers assume EV-optimal play, the same line the trainer teaches.
Are these numbers simulated?
No. They come from exact enumeration: every one of the 2,598,960 possible deals, played optimally across all 32 holds, with the resulting category probabilities summed. The probabilities add to 1 to nine decimal places and reproduce the 99.5439% return independently.