Contents of this section:
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Here's what you'll find in this section:
Probability provides a mathematical description of randomness. A phenomenon
is called random if the outcome of an experiment is uncertain. However,
random phenomena often follow recognizable patterns. This long-run regularity of
random phenomena can be described mathematically. The mathematical study of
randomness is called probability theory.
 | Experiments
and Events
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 | A well-defined procedure resulting in an outcome, e.g., rolling a die,
tossing a coin, dealing cards.
| experiment An experiment with the following characteristics:
- The set of all possible outcomes is known before the
experiment.
- The outcome of the experiment is not known beforehand.
| Space The set of all possible outcomes of the experiment. We use S
to denote the sample space.
| Any subset of the sample space. |
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| Properties
of Probability
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| Some
Set Theory
| Conditional
Probability and Independence
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EXAMPLE: Let S be a deck of
52 cards (4 suits, 13 cards per suit), A be the event that a king is
drawn, and B be the event that a spade suit ( 
) is drawn.
Then
Thus, A and B are independent.
NOTE: From conditional probability,
we can get the following relationship:
| The
Equally Likely Case
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In general, the probability 
of any event A is found by summing the individual probabilities P(X=x)
for all possible outcomes x in A. For events with equally
likely outcomes, probabilities are very simple to compute: if a random
phenomenon has k possible outcomes, all equally likely, then each
individual outcome has probability 1/k. That is,
| Relative
Frequency Interpretation of Probability
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Recall that we defined the probability of event A as
This is called the classical probability definition. Another way
to interpret probability is as the long-run relative frequency (long-run
fraction) of the event. That is, if I flip a fair coin hundreds and
hundreds of times, the fraction of heads will be very close to 0.5. The more
I repeat the experiment, the closer to 0.5 the relative frequency will be.
This is the same result the classical definition gives us. The relative
frequency interpretation of probability works especially well for repeatable
events, e.g., flipping a coin, rolling dice, drawing cards, etc. See the
``Relative frequency interpretation of probability'' concept lab for an
example.
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Internet References
Related pages in this website
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. That is, knowing that event B occurred gives us no additional
information on the probability that event A occurs.
, then A and B are dependent.
