Difference between revisions of "Expected value"
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More formally, we can define expected value as follows: if we have an event <math>Z</math> whose outcomes have a [[discrete]] [[probability distribution]], the expected value <math>E(Z) = \sum_z P(z) \cdot z</math> where the sum is over all outcomes <math>z</math> and <math>P(z)</math> is the probability of that particular outcome. If the event <math>Z</math> has a [[continuous]] probability distribution, then <math>E(Z) = \int_z P(z)\cdot z\ dz</math>. | More formally, we can define expected value as follows: if we have an event <math>Z</math> whose outcomes have a [[discrete]] [[probability distribution]], the expected value <math>E(Z) = \sum_z P(z) \cdot z</math> where the sum is over all outcomes <math>z</math> and <math>P(z)</math> is the probability of that particular outcome. If the event <math>Z</math> has a [[continuous]] probability distribution, then <math>E(Z) = \int_z P(z)\cdot z\ dz</math>. | ||
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| + | == Uses == | ||
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| + | *Expected value can be used to, for example, determine the price for playing a probability based carnival game. You first find the expected value per player. Then you can charge a reasonable price so that you gain money, but the price isn't unreasonably high. | ||
== Example Problems == | == Example Problems == | ||
Revision as of 12:48, 19 January 2008
| This is an AoPSWiki Word of the Week for Jan 17-23 |
Given an event with a variety of different possible outcomes, the expected value is what one should expect to be the average outcome if the event were to be repeated many times. Note that this is not the same as the "most likely outcome."
For example, flipping a fair coin has two possible outcomes, heads (denoted here by 
) or tails (
). If we flip a fair coin repeatedly, we expect that we will get about the same number of heads as tails, or half as many as the total number of flips. Thus, the average outcome is 
. Note that not only is this not the most likely outcome, it is not even a possible outcome for a single flip.
More formally, we can define expected value as follows: if we have an event 
whose outcomes have a discrete probability distribution, the expected value 
where the sum is over all outcomes 
and 
is the probability of that particular outcome. If the event has a continuous probability distribution, then 
.
Uses
- Expected value can be used to, for example, determine the price for playing a probability based carnival game. You first find the expected value per player. Then you can charge a reasonable price so that you gain money, but the price isn't unreasonably high.
Example Problems
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