The math kids use to make real decisions.

The three, by a lot.

1. Picture 100 of each shot.
2. 100 long twos: 40 go in. 40 × 2 = 80 points.
3. 100 threes: 36 go in. 36 × 3 = 108 points.
4. Threes miss more, yet score 28 more points per 100 shots.

The idea: expected value. Chance × payoff tells you what a choice is worth on average. It’s why threes are now 42% of NBA shots and long twos only 5%.

2024–25 league averages from Basketball-Reference. Long two = 16 feet out to the 3-point line.

Too lucky to believe.

1. 1 in 20 is 5%. 5% of 262 trades is about 13 pearl trades.
2. He got 42. That’s more than 3 times what you’d expect.
3. Small surprises happen all the time. Huge ones, again and again, almost never do.
4. Speedrun moderators worked out the odds of his pearl and blaze rod luck together: 1 in 7.5 trillion. They removed the run.

The idea: expected count. Multiply the chance by the number of tries to see what “normal” looks like. The further a result lands from normal, the more it needs explaining.

Numbers from the moderators’ December 2020 report, via Defector.

The 6. It pays 2½ times as often.

1. Two dice can land 36 ways (6 × 6).
2. A 6 can be rolled 5 ways: 1+5, 2+4, 3+3, 4+2, 5+1.
3. A 2 has 1 way (1+1). A 12 has 1 way (6+6). Together: 2 ways.
4. 5 ways beats 2 ways. 5 ÷ 2 = 2½.

The idea: counting outcomes. When every outcome is equally likely, the chance is the ways it can happen ÷ all the ways. The dots on Catan’s number tokens count those ways for you.