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Probability and Expected Value

Independent Events

If two random events are independent of one another, the probability that both will occur is the product of the probabilities of the individual events.

For example, if I flip a coin, the probability that it will land heads up is 0.50 (50%). If I roll a six sided die (the usual kind found in board games), the chance that it will stop with a 6 on top is 1/6 or about 0.17 (17%). If I do both at the same time, the chance that the coin will be heads up AND the die will stop with 6 up are 0.5 * 0.17 = .085 (8.5%). By extension, the chance that something else will happen is 1-0.085 or 0.915 (91.5%).

Expected Value

The expected value of an uncertain event is the sum of the possible payoffs multiplied by each payoff's chance of occurring.

Suppose I offer you the following gamble: I roll a six sided die and give you $6 if a 6 comes up, $2 if a 3, 4 or 5 comes up, and nothing otherwise. Since there is a 1/6 chance of each number coming up, the outcomes, probabilities and payoffs look like this:

Outcome Probability Payoff
1 1/6 $0
2 1/6 $0
3 1/6 $2
4 1/6 $2
5 1/6 $2
6 1/6 $6

The expected value is sum of the entries in the last two columns multiplied together:

(1/6)*0 + (1/6)*0 + (1/6)*2 + (1/6)*2 + (1/6)*2 + (1/6)*6 = $2.

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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 08/17/2016