Bad beats: how often they should happen

All-in before the flop with aces, against kings.

Aces have 81.95% equity, so kings take about 18% of the pot on average. That is close to one hand in five. Get aces against kings all-in five times and the most likely single outcome is that you lose one of them.

Neither player could reasonably fold, so the trainer's glossary calls this a cooler rather than a bad beat. Either way, it was not a mistake by the player with aces.

On the river the sum is simple. Four board cards and both players' hands are known, so 44 cards are unseen. Your opponent's chance is their outs divided by 44:

A two-outer on the river comes in about one time in 22. If you get your money in on the turn as a 21-to-1 favourite often enough, you will see it regularly, and each one will feel like it should not have happened.

Two players each hold a pocket pair and both flop a set. The smaller set usually loses, and it loses a big pot it could not fold.

With both hands known, 48 cards are unseen and there are 17,296 possible flops. Your pair flops a set or better whenever at least one of the two cards left of your rank comes:

For both players to flop a set or better, the flop needs at least one card of each rank. Count them: one of each with any third card (2 × 2 × 44 = 176), or two of one rank and one of the other (2 + 2 = 4).

The 180 includes 4 flops where one player makes quads. Strictly set against set is 176 of 17,296 flops, about 1.02%, conditional on the two players holding different pocket pairs.

Three bad beats in an evening can feel personal. A binomial calculation shows how often to expect a run like that.

Say you get all-in ten times in a session, each time as an 80% favourite. Assume independent all-ins with no ties and an 80% win probability each time. The number you lose then follows a binomial distribution:

Win all ten

10.7%

0.8 to the power of 10

Lose at least one

89.3%

nearly every session

Lose three or more

32.2%

about one session in three

Losing three or more of ten all-ins as a big favourite happens in about one session in three. That result is common under the stated assumptions. On its own, it is not evidence that something is wrong with the cards or your play.

A bad beat costs you a pot you were favoured to win. It costs more if it changes how you play the next hands, which is tilt. The chips in that pot are gone either way. Your next decisions are the part you can still get right.

The trainer tracks luck and decisions separately. At the cash table, a session's result is split into luck and play: what the all-in cards did against your equity, and what your graded decisions cost. A hand you got in ahead and lost shows as bad luck, not a bad decision. A cooler session at the cash table deals you a few hands you cannot avoid losing, then grades how you play the hands after them.