Instrument 3 · expected value

What a ticket is worth

Multiply each outcome by what it pays, add them up. That is the whole formula, and it is the only one on this list you can use in a shop. What it does not say is whether you should play — because a positive expectation and a short bankroll are different questions.

What the bet is worth, by chance of winning

Two hundred plays, five times over

expected value break-even — the fair price a run a run that went broke

The same statement, with the numbers in it

E = p \cdot win - (1 - p) \cdot stake

0.00 = 0.30 \cdot 10 - (1 - 0.30) \cdot 2

The fair price is where this is zero: p·win = (1−p)·stake. Everything a bookmaker, an insurer or a lottery does is set the price on the wrong side of that line — and everything you do is notice by how much.

Fine tune

Chance of winning 30%

What winning pays 10

What it costs to play 2

Your bankroll 20