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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 longrun regularity of random phenomena can be described mathematically. The mathematical study of randomness is called probability theory.
Conditional probability reduces the sample space we have to work with.
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:
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,
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 longrun relative frequency (longrun 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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