Calculation of the Expectation
To calculate the expected value, one must know the probabilities and payoffs of all possible outcomes give what has already transpired. The EV is the sum over all possible outcomes of the product of the payoff of the outcome and the probability of that outcome.
EV = P(outcome 1) * Payoff 1 + P(Outcome 2) * Payoff 2 + ... + P(Outcome n) * Payoff n
Two players toss a fair coin. If the coin falls heads up, player B pays player A a dollar. On tails, A pays B a dollar. The probability of either heads or tails is 50%. Hence the EV for both players is
EV = 50% * (+1$) + 50% * (-1$) = 0$
Another method of calculation for finding the profit of a bet is:
Players A and B toss a fair coin. The payoffs are as before, except that at the beginning of the round each puts a dollar in the pot and the winner takes the pot at the end. Each player wins the pot of $2 50% of the time. Their pot equity is therefore 50% of $2 or $1. Hence, the EV isEV = Equity - Bet = 1$ - 1$ = 0$
Neither player wins or loses anything since they lose a dollar half the time and win it back the other time.