Expected value

Pronunciation: /ɪkˈspɛkt ɛd ˈvæl yu/ ?

In probability, an expected value is the probability of each and every event in a sample space times the probability that the event will occur.[1] Since the expected value is numeric, the outcomes of the experiment must also be numeric. For experiments with finite outcomes, this can be calculated as the sum of the values of each outcome times the probability of that outcome occurring.

Example 1: Role of one die

Table 1 shows the outcomes and their probabilities for the roll of one fair die.

Outcome
of roll
ProbabilityOutcome ×
Probability
11/61*(1/6)=1/6
21/62*(1/6)=2/6
31/63*(1/6)=3/6
41/64*(1/6)=4/6
51/65*(1/6)=5/6
61/66*(1/6)=6/6
Table 1: Expected value of role of one fair die

So, the expected value is (1/6)+(2/6)+(3/6)+(4/6)+(5/6)+(6/6)=3.5.

Example 2: Roll of two dice

Table 2 shows the outcomes and their probabilities for the roll of two fair dice.

Outcome
of roll
ProbabilityOutcome ×
Probability
21/362*(1/36)=2/36
32/363*(2/36)=6/36
43/364*(3/36)=12/36
54/365*(4/36)=20/36
65/366*(5/36)=30/36
76/367*(6/36)=42/36
85/368*(5/36)=40/36
94/369*(4/36)=36/36
103/3610*(3/36)=30/36
112/3611*(2/36)=22/36
121/3612*(1/36)=12/36
Table 2: Expected value of role of two fair dice

So, the expected value is (2/36)+(6/36)+(12/36)+(20/36)+(30/36)+(42/36)+(40/36)+(36/36)+(30/36)+(22/36)+(12/36)=(252/36)=7.

Cite this article as:


Expected value. 2010-02-02. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/expectedvalue.html.

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2010-02-02: Added "References" (McAdams, David.)
2009-12-01: Initial version. (McAdams, David.)

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