In discussing Three Card Poker, I've often mentioned that the house edge on the most common Pair Plus pay table is 7.28 percent, while the original version had a 2.14 percent edge.
The difference is the payoff on flushes. When Derek Webb invented the game, it paid 4-1 on flushes. Today's most common version drops that by a unit to 3-1.
All that brought a inquiry from Marcus, a reader and Three Card Poker Player.
"Can you explain how that works?" he asked. "Where does the house edge come from, and why does that small change make such a big difference."
The house edge in all casino games comes from paying less than the true odds of winning. Losing bets are the ones that feed the house coffers, of course, but if winners were paid at true odds, the house would pay out as much to winners as it collects from losers. By paying less than true odds, it pays less to winners than it collects from losers.
In Three Card Poker, there are 22,100 possible three-card hands in which card order makes no difference.
If you have a straight consisting of 9 of hearts, 10 of clubs and Jack of spades, your payoff is the same regardless of whether the cards appear in the order of 9-10-Jack, 9-Jack-10, 10-9-Jack, 10-Jack-9, Jack 10-9 or Jack 9-10. So in calculating the number of possibilities, those same cards in different orders are counted as one hand.
Imagine you wagered $10 per hand for 22,100 hands, and got each possible hand once. Your total risk would be $221,000.
Given the common pay table of 40-1 on a straight flush, 30-1 on three of a kind, 6-1 on a straight, 3-1 on a flush and 1-1 on a pair, these would be your returns:
On each of the 48 possible straight flushes, you would keep your $10 wager and get $400 in winnings, for a total of $19,680.
On each of 52 possible four of a kinds, you'd keep the $10 wager and get $300 in winnings, for a total of $16,120.
On each of 720 possible straights, you'd keep the $10 wager and get $60 in winnings, for a total of $50,400.
On each of 1,096 possible flushes, you'd keep the $10 wager and get $30 in winnings, for a total of $43,840.
On each of 3,744 possible pairs, you'd keep the $10 wager and get $10 in winnings, for a total of $74,880.
On the other 16,440 possible hands, you lose your money.
Add up the totals from each of the winning hands and you get $204,920. When you subtract that from the $221,000 you've wagered, you see the house has kept $16,080.
Divide your $16,080 in losses by the $221,000 in wagers, then multiply by 100 to convert to percent, and you get 7.28 percent. That's the house edge.
What if flushes paid the original 4-1 instead of 3-1? Then, given $10 wagers, the extra $10 on each of the 1,096 flushes would bring the total return on those hands to $54,800. That would raise your overall payback to $215,880, and lower the house profit to $5,120.
If you then divided $5,120 in losses by $221,000 in wagers, then multiplied by 100 to convert to percent, the result would be 2.32 percent — the original house edge on Pair Plus.
That's the kind of thing video poker players have faced for decades. A small change in payoff on hands low on the pay table make big differences in your return.
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