Reading the maths
An average is not a session forecast
Expected value describes a mathematical average. It does not describe the money a particular person will finish with.
The average over possibilities
An expected value weights possible outcomes by their probabilities. For example, a hypothetical event paying £2 with probability one quarter and £0 otherwise has an expected gross payout of 50p. If each attempt costs £1, the expected net result is a loss of 50p per attempt. Neither possible gross payout is actually 50p.
Describe the whole model
Write down whether a payout includes the returned stake, what each attempt costs and whether the probabilities remain fixed. These details change the calculation. A precise-looking number is unhelpful if readers cannot reproduce its assumptions.
Use the distinction
A negative expected result is not proof that every short sequence loses. Equally, a favourable short sequence does not change the underlying model. Casino mathematics is useful for recognising uncertainty and cost; it cannot make gambling income dependable. Our examples are educational calculations, not estimates for an identified operator or game.