Play the games, see the math

Every slot result is random, but the reel strips are weighted. This machine has 32 stops per reel and only two 7s, so three 7s come up once every 4,096 spins on average. The RTP shown is computed exactly from all 32,768 reel combinations.

Near misses. Two 7s and a miss looks “close”, but it is the same loss as any other. Near misses have been shown to raise the urge to keep playing [source].

Losses disguised as wins. A single ● pays 1 credit on a 2-credit bet. The machine treats it as a payout, yet you are down 1. Research on multi-line slots found these outcomes are experienced much like real wins [source].

This table’s edge was measured by simulating 1,000,000 hands of its exact rules with basic strategy: about 0.34% of the starting bet, with a simulation margin of roughly ±0.25 points. That is close to the lowest edge in a casino, but only with perfect basic strategy. Hunches, never busting or copying the dealer cost several times more.

Why card counting isn’t a plan. Counting only shifts the odds a little, and only some of the time. It needs flawless strategy and counting under pressure, plus a large bankroll to ride out normal swings. Casinos can also shuffle early, change the rules or ask counters to leave. For almost everyone, the edge stays with the house.

The wheel has 38 pockets: 1–36 plus 0 and 00. A single number pays 35 to 1, but the fair price would be 37 to 1. Bet 1 credit on each of the 38 pockets and you get back 36: the missing 2 of 38 is the house edge, 5.26%, and it is the same on every bet on this table.

The gambler’s fallacy. After five reds in a row, black feels “due”. It isn’t: the wheel has no memory, so each spin has the same odds. People tend to expect short runs to look like the long-run average, a bias described by Tversky and Kahneman [source].

This is a casino-style hold’em: you play against the dealer, not other players, so the house wins through the rules. The dealer only “qualifies” with a pair of 4s or better, and the payouts are set so the house keeps a share of every ante. We measured this table by simulating 250,000 hands with a simple strategy (call with any pair or better, an ace or king in hand, or four to a flush or straight; otherwise fold): the house kept about 6.05% of each ante. Expert strategy lowers that, but never to zero.

Edge vs rake. In a poker room you play other people and the house has no edge; instead it takes a rake, a small cut of each pot or a seat fee. Either way, the money flowing through the table shrinks over time, and only players who beat the others by more than the rake come out ahead.

You split seven cards into a 5-card hand and a 2-card hand; the 5-card hand must be the stronger one. Most hands push, which makes the game slow and steady, but the house still keeps an edge. We measured it by simulating 250,000 hands with you and the dealer both setting the house way: about 2.71% per hand. Two things create it: the 5% commission on every win, and the dealer winning exact ties (“copies”).

The “house way” is the fixed method dealers use to set hands: for example, split two pairs unless you hold an ace and the high pair is 10s or lower, and put the second-best pair of a full house in front. Following it is close to the best you can do, so there is little room to beat the game by skill.

Craps shows the spread between good and bad bets better than any other game. The pass and don’t pass lines cost about 1.4%, and the odds bet behind them pays at true odds, so it has no edge at all. The one-roll bets in the middle cost 9% to almost 17%.

Every edge here comes from counting the 36 ways two dice can land. Any 7 wins on 6 of 36 rolls but pays 4 to 1: per 36 credits bet, you get back 6 × 5 = 30, a 6/36 = 16.67% edge. Any craps (2, 3 or 12) wins 4 times in 36 and returns 4 × 8 = 32, a 4/36 = 11.11% edge. Eleven wins 2 times and returns 2 × 16 = 32, also 11.11%. The field wins on 16 of 36 rolls, with 2 and 12 paying double, so it returns 34 of 36, a 5.56% edge. Pass line works out to 244/495 wins, an edge of 7/495 = 1.41%.

Place bets pay a little less than true odds. A 6 or 8 rolls 5 ways against 6 ways for a 7, so true odds are 6 to 5, but it pays 7 to 6: an edge of 1/66 = 1.52%. The 5 and 9 (4 ways vs 6) pay 7 to 5 instead of 3 to 2, a 4.00% edge. The 4 and 10 (3 ways vs 6) pay 9 to 5 instead of 2 to 1, a 6.67% edge. Aces and twelve each roll 1 way in 36 and pay 30 to 1, returning 31 of 36: a 13.89% edge. Ace-deuce rolls 2 ways and pays 15 to 1, returning 32 of 36: 11.11%. The horn splits the bet over 2, 3, 11 and 12: 12.50%. Hard 4 and hard 10 win 1 way and lose 8 ways (any 7 or the easy number) at 7 to 1: 11.11%. Hard 6 and hard 8 win 1 way, lose 10 ways, at 9 to 1: 9.09%. Come and don’t come work like pass and don’t pass, starting from the next roll.