Poker and games

The pair of twos that beats two big cards

Two players go all in before the flop. One holds a pair of twos, the other holds ace king of different suits. Card for card the ace king looks far stronger, and it wins less often.

The twos are already a pair. Ace king has to pair up to pass them, and it has six cards to catch across five community cards. Deal out every possible board and the twos take it about 52 times in 100, a figure you can check on a hand equity calculator such as CardFight. That is Pocket twos against ace king, and it is the cleanest example of a made hand beating two chances.

Nothing here is advice about how to play. It is a probability, and probabilities are countable.

Why does chance feel wrong at a table?

Three mechanisms cover most of the misreadings, and none of them need advanced maths.

Pairs grow much faster than people. Ask how many people you need in a room before two of them share a birthday and most answers land near 183, half of 365. The right answer is 23. Twenty three people form 253 pairs, and the chance that all 253 pairs miss each other falls just under half, so a match turns up 50.7 percent of the time. Counting people gives one number, counting pairs gives another, and the question is about pairs. How many people share a birthday is the whole thing in one card.

Equipment keeps no memory. A European wheel has 37 pockets, 18 of them red. After nine reds in a row the chance of red on the next spin is still 18 out of 37, which is 48.6 percent. The wheel does not rearrange its pockets to settle a debt. Nine reds in a row is unlikely in advance and ordinary once it has happened, and those are separate statements.

One outcome is not a class of outcomes. A lottery line of 1, 2, 3, 4, 5, 6 has exactly the same chance as any scattered set of six numbers. Scattered sets look random because there are far more of them, and that fact is about the whole list of possible draws rather than about your ticket. Playing 1, 2, 3, 4, 5, 6 separates the two questions.

How do you count a probability in your head?

Four moves handle most of what comes up in a game.

Count what has to disappear. A knockout bracket starts with 128 players and one loss sends you home. Every game removes exactly one player, and 127 players have to go before one remains, so the bracket needs 127 games. Adding the rounds gets there too, at seven times the effort: 64, 32, 16, 8, 4, 2 and 1. Games in a 128 player bracket rewards the one step version.

Multiply outs by four with two cards to come, and by two with one. Nine hearts left to finish a flush after the flop: nine times four is 36, and the exact figure is 35 percent. One card to come: nine times two is 18, against an exact 19.6. The shortcut runs a point or two off and it is quick enough to use while the hand is live. Nine outs, two cards to come is that estimate under a timer of your own.

Count the complement when the direct count is long. The chance of at least one match is easier to reach through the chance of no match at all. This is what makes the birthday count tractable, and it works anywhere the word at least appears in the question.

Round hard, then take the fraction once. A card room runs 22 tables of nine seats with roughly seven seats in ten filled. Round 198 seats to 200, take seven tenths, and you have 140 against a true 138.6. How full is the card room accepts anything in range, because the value of the estimate is in the method.

Where the counting usually breaks

Stopping at the count instead of the amount. Seven stacks of 20 chips is 140 chips, which is a count. At 1.50 a chip it is 210 dollars, and against a 200 dollar buy in the night finished 10 up. Counting down the stack catches the missing multiplication.

Charging a small edge on the wrong total. A 200 dollar bonus that requires 20 times the bonus to be wagered on a game returning 95 cents per dollar puts 4,000 dollars through the house. Five cents of every dollar stays there, and 5 percent of 4,000 is exactly 200. The edge sounds tiny beside the gift, and it is charged on a total twenty times larger, so the expected survival of the bonus is nothing at all. The 200 dollar bonus with strings is a wagering requirement written out as arithmetic.

Reading a rare event as an overdue one. Runs happen. A sequence that would be surprising to predict is unremarkable to observe after the fact, and the next trial starts from the same pockets, the same deck and the same odds.

Answering the wrong question. How many people give a better than even chance of a shared birthday, and how many people give a better than even chance that someone shares yours, are different questions with very different answers. Read which one is on the card.

The syndicate that bought every ticket

In 1992 the Virginia lottery drew six numbers from 44, which gives 7,059,052 possible tickets. At a dollar a line, covering every combination cost about a quarter of a jackpot that had passed 27 million dollars.

An Australian syndicate set out to buy the lot. It filed more than five million tickets in a few days, ran out of time before it finished, and won anyway. The episode was reported at the time by UPI.

The arithmetic only works while the prize sits far above the cost of the cover, and a second winner splitting the jackpot would have sunk it. Buying every ticket in Virginia is the card version, and it is a story about a payout table rather than a system for beating one.

Where this shows up away from a table

Sports brackets, where the number of games is fixed before anyone plays. Fantasy drafts and raffles, where one specific outcome gets confused with a class of them. Insurance quotes and warranty offers, where a small percentage is charged against a large base. Any promotion with a wagering requirement, a minimum spend or a qualifying period, where the terms decide the arithmetic long before luck does.

The habit worth keeping is narrow. Before you accept a chance, ask what is being counted: people or pairs, one ticket or every ticket, the count of chips or the money they represent.

Answer them one at a time

The feed above runs these as cards. A situation, one question, your answer, and a line underneath naming the count that gets there. Nothing is timed and nothing is scored.

The reading version of the same material lives on the blog, starting with the Monty Hall problem, which is the classic case of an answer that stays wrong until you count it. For the money side, Everyday money takes offers, fees and receipts.