Expected Value

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Expected Value

The expected value is the long-term average of an experiment.

Example 1:

In a Roulette game, bet $1.00 on 0 and 00. Either win $17 if 0 or 00 comes up, or lose $1.00 if any other number comes up. However, to determine the Expected Value, the long-term outlook of the bet, the probability of each outcome must be considered.

$17(2/38) + -$1(36/38) = $34 - $36
18

= - $2 = - $0.052 ~ - 5 cents every bet.
18

That’s why the house wins. They are there for the long-term, so with every bet, they will make 5 cents.

Rule: to find expected value, multiply value of each outcome by its probability and add the results.

Decision Theory: What is the better bet, $1 on a single number or $1 in the lottery?

Expected Value of Roulette bet

= $35(1/38) + (-$1)(37/38) = 35/38 – 37/38

= -$2/38 = - .052 ~ -5 cents.

(same as EX 1)

Expected value of lottery 53/6 with $6,000,000 jackpot

= $6,000,000(1/22,957,480) + (-$1)(22,957,479/22,957,480)

= ( 6,000,000 – 22,957,479)/22,957,480

= -$16,957,479/22,957,480 = -$.7386 ~ - 74 cents.

the roulette bet is – 5 cents, while the lottery is – 74 cents.

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