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PROBABILITY THEORY FOR EE 150
Background for coin toss problems and chemical reaction simulations
Probability theory is the branch of mathematics concerned with analysis of random phenomena.
The central objects of probability theory are random variables, stochastic processes, and events:
mathematical abstractions of nondeterministic events or measured quantities that may either be
single occurrences or evolve over time in an apparently random fashion. Although an individual
coin toss or the roll of a die is a random event, if repeated many times the sequence of random
events will exhibit certain statistical patterns, which can be studied and predicted. Two
representative mathematical results describing such patterns are the law of large numbers and the
central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human
activities that involve quantitative analysis of large sets of data. Methods of probability theory
also apply to description of complex systems given only partial knowledge of their state, as in
statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature
of physical phenomena at atomic scales, described in quantum mechanics.
In probability theory, an event is a set of outcomes (a subset of the of a larger set which is usually
defined as the set of possible outcomes of interest ) to which a probability is assigned. Typically,
when the set of possible outcomes is finite, any subset of the set of possible outcomes is an event
(
i
.
e
. all elements of the of the set of possible outcomes are defined as events).
Probability plays a role in many sectors of modern life. Most of us play games of chance in some
way or other. This may vary from a simple board game to playing games in a casino for money.
In corporate life, actuaries are employed to assess risk of almost every kind so that insurance
companies make profits on the premiums they set.
There are a number of ways to try to determine the probability of an event. Here we use
probability trees. This method is useful when we have basic activities with known theoretical
probabilities, such as rolling a dice, tossing a coin, drawing a card from a deck or taking a ball
from a barrel. When these activities are repeated we can find the probability of an event such as
getting three sixes in three rolls of a dice, by constructing a probability tree.
Probability trees can also be used
in situations where two different basic events are combined
and to compute conditional probability. But first we will just be concerned with learning how to
construct and use probability trees when one activity is repeated.
Along the way one should always look for patterns in your work. Here we will look for
generalizations. What will happen when a coin is tossed n times? What is the likelihood of getting
precisely one Tail?
Two? These generalizations are basically like finding algebraic patterns but
involve slightly more sophistication.
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 Fall '08
 Dr.Burke
 Probability, Probability theory, Playing card, Coin flipping

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