KALexpected valueof a discrete random variable Y is defined as follows: E[Y] = Σ y×p(y), where the summation (Σ) is over all possible values y that the random variable Y can assign to the possible outcomes times p(y), the probability associated with that value. In gambling games, however, it may be more sensible to consider only those events that have a reasonable chance of happening. Thus we may define the
reasonable expected valueof a discrete random variable Y as follows: E/e[Y] = Σ y×p(y), where the summation is over all values of Y for which p(y) > e, where e is some very small, i.e. highly unlikely, probability.
For example, in the Powerball Lottery the reasonable expectation
would be as listed for say e = 1/100,000,000, namely 19.7 cents for a $1 bet (or 34.5 cents per dollar if one uses the Power Play option). While if e were chosen to be 1/3,000,000 the event of 5W 0R would also be eliminated (since its probability of happening is 1/3,563,609), yielding a reasonable expectation
of only .141 or 14.1 cents for a $1 bet (or 24.7 cents per dollar if one uses the Power Play option).
reasonable expectationin Hot Lotto would be as stated, namely 23.7 cents for a $1 bet (or 35.6 cents per dollar if one uses the Sizzler option).
reasonable expectationin Gopher5 would be as stated, namely 22.75 cents for a $1 bet.
Scott Adams