# 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.

URL: https://wilcoxen.maxwell.insightworks.com/pages/126.html
Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 08/22/2018