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The author expects you to have a profound knowledge of linear algebra, analytic geometry, and calculus.
Forewarned is forearmed.
The purpose of probability theory is to build up a mathematical framework for various random experiments. Yeah, there will be a bunch of formulas to construct all that mathematical grounds.
A random experiment is a procedure whose collection of every possible outcome can be described before its performance and can be repeated under the same conditions.
A sample is an outcome of a random experiment. A sample space is a collection of samples:
An event is a subset of a sample space. An event space is a collection of events, which complies with the sigma-algebra axioms:
I. an event space contains an entire sample space:
II. an event space is closed under complementation:
III. an event space is closed under countable unions:
Mutually exclusive events are those which can’t happen at the same time:
Collectively exhaustive events are those the union of which is the sample space:
A partition of an event space is such events which are collectively exhaustive and mutually exclusive simultaneously:
A probability measure is a function from an event space into the real numbers, which complies with Kolmogorov’s axioms below:
I. the probability measure of an entire sample space is equal to 1:
II. the probability measure of an event is greater than or equal to 0:
III. the probability measure of a union of mutually exclusive events is equal to the sum of probabilities of events:
There are some properties you need to keep in mind as a result of the axioms above:
I. the sum of the probability of an event and the probability of its complement is equal to 1:
II. the probability of a null event is equal to 0:
III. the probability of an event is greater than or equal to the probability of its subevent and less than or equal to the probability of its superevent:
IV. the probability of an event is greater than or equal to 0 and less than or equal to 1:
V. the probability of a union of two events is equal to the sum of the probability of the first event and the probability of the second, subtracting the probability of their intersection:
VI. the probability of a difference between two events is equal to the difference between the probability of the first event and the probability of their intersection:
A probability space is a triplet, consisting of a sample space, an event space, and a probability measure:
There are two types of probabilities:
I. A classical probability of an event (a discrete case) is the number of samples in that event divided into the total number of samples:
II. A geometric probability of an event (a continuous case) is the area of samples in that event divided into the total area of samples:
Let’s utilize that knowledge and apply that framework to some examples from real life.
A discrete example. We cast dice. The question is: what are the probabilities of a) both equal and even; b) both equal and odd; c) their union; d) their intersection?
In this case, casting dice is a random experiment, which means we should define a probability space:
so that we can calculate the probabilities:
A continuous example. We toss darts. The question is: what are the probabilities of the dart landing in the a) red circle (0 < r ≤ 2); b) blue circle (2 < r ≤ 4)?
In this case, tossing darts is a random experiment, which means we should define a probability space:
so that we can calculate the probabilities:
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