Why the royal flush is worth two percentage points of the return

Return is a sum, not a single fact. Multiply every payout by how often correct play reaches it, add the lot up, and you get the number at the bottom of the paytable. For 9/6 Jacks or Better that number is 99.54%. What the number hides is how unevenly the contributions are spread, and the royal flush row is the extreme case: it is 800 coins per coin bet, arriving once in about 40,400 hands.

800 × (1 ÷ 40,400) = 0.0198. Just under two percentage points.

The return, row by row

The royal contributes more than the straight flush and less than everything else, which sounds unremarkable until you put it beside the 8/5 machine. Trimming the full house from 9 to 8 and the flush from 6 to 5 costs about 2.2 points and turns a full-pay machine into a mediocre one. The royal row is worth nearly as much as that entire downgrade — and unlike the full house, you will collect it roughly never.

The machine you actually play

Strike the royal out and the other rows sum to 97.56%. That is the game as experienced by almost everyone who sits at it: a 97.6% machine with a lottery ticket attached. The advertised 99.54% is an accounting fact about a very long run, and about two of its points are held in escrow against a hand that arrives, on average, once every 101 hours at a brisk 400 hands an hour.

The waiting times are worth stating plainly, because "once in 40,400" reads as more frequent than it behaves:

• Over 10,000 hands — a serious month of play — the chance of at least one royal is 22%.
• The median wait is about 28,000 hands. Half of all players get there sooner; half never do.
• Play exactly 40,400 hands and you still have a 37% chance of having seen none.

The same row is also the source of most of the variance: remove it and the standard deviation falls from 4.42 to 1.93. Two points of return and roughly three quarters of the turbulence, from one line of the paytable.

Why the five-coin rule exists at all

The royal is the only row on a casino paytable that does not multiply cleanly. At one to four coins it pays 250 per coin; at five it pays 800. The 1.98 points above assume the 800 rate. At the 250 rate the same frequency yields 0.62 points, so short-coin play throws away about 1.36 points before a single card is dealt — roughly 98.2% instead of 99.54%, and a little better than that once you re-optimise your holds for a smaller top prize.

What it changes about how you play

Two points of return concentrated in a rare hand pulls strategy in a specific direction: chase it in a few clearly defined spots, and ignore it everywhere else.

Hold four to a royal and you break a made flush to do it. That looks reckless and is not. Say you hold A♠K♠Q♠J♠ plus one more spade — a completed flush worth 6 per coin. Discard the fifth spade and 47 cards remain: one is the T♠ and pays 800, seven are spades and pay 6, three are offsuit tens and make a straight worth 4, and twelve pair one of your high cards for 1. That averages (800 + 42 + 12 + 12) ÷ 47 ≈ 18.4 per coin. Three times the made flush, and the royal card supplies 17 of those 18.4.

The line has to be drawn somewhere, though, and the standard hold order draws it tightly:

1. Four to a royal beats a made flush, a made straight and three of a kind — but never a made straight flush or four of a kind, which already pay more than the draw is worth.
2. Three to a royal beats a low pair and four to a flush, and loses to a high pair. One extra card to find makes all the difference; the draw is worth roughly a coin and a half.
3. Two to a royal is not a hand. Hold the high cards if the rest of the chart says so, not because they happen to share a suit and a picture.

That is the entire influence of two percentage points on your decisions: a handful of holds, a few times an hour. Everything else you do at the machine is decided by two pair, trips and high pairs — the rows carrying 70 points of the return between them.