Expected Values and Fair Games

For a discrete random variable, \(E(X)=\sum\limits_x xP(X=x)\). This is a probability-weighted long-run average; it need not be a possible outcome.

For this prize distribution, \(E(X)=0(0.5)+2(0.3)+5(0.2)=1.6\). A fair entry fee is \(1.6\) units of money: expected net gain is zero. Distinguish a gross prize from net gain after paying the fee.

If an event has probability \(p\) on each of \(n\) trials, its expected number of occurrences is \(np\). For \(80\) deliveries each with delay probability \(0.03\), the expected number delayed is \(2.4\); this is not a claim that a realised count can be fractional.

Keep winnings and profit distinct. If \(W\) is the prize received and the entry fee is \(c\), the net gain is \(W-c\). A fair fee satisfies \(c=E(W)\); a fair game has expected net gain zero, not necessarily an equal chance of winning and losing.

An expected value can lie between the possible values of a discrete variable. It describes a long-run average over repetitions, not a guaranteed result of one play. Check that the units of the expected value match the variable.