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  Theory of Probability

Probability:

Basic Definitions:

Experiment: Any happening whose outcome or result is uncertain.

Outcomes: Possible outcomes/results from an experiment.

Sample Space: It is the set of all possible outcomes.

Event: It is the subset of sample space. There will be one or more outcomes.

Equally Likely Events: The events that have similar chance of occurring.

Probability: It is a chance that an event will take place. Theoretically for equally likely events, this is the number of ways an event can take place divided by number of outcomes in sample space. Empirically, it is the long term relative frequency.

Independent Events: It is the events in which the occurrence of one event doesn’t change the probability of occurrence of the other. One doesn’t influence the other.

Dependent Events: The events which are not independent.

Mutually Exclusive Events: It is the events which cannot occur at the same time. These are disjoint events.

All Inclusive Events: The events whose union includes the totality of sample space.

Complementary Events: Two mutually exclusive events which are all inclusive.

Sample Spaces:

The sample space is the set of all possible outcomes in experiment and is represented by the capital letter S.

When we were to roll a single die, then S = {1, 2, 3, 4, 5, 6}, that is, the set of all possible outcomes.

When we were to roll two dice and look at the sum of two dice, then S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

Equally Likely Events:

Though, not all the sample spaces are created uniformly. However, that last illustration is not. There is just one way that a sum of 2 can be rolled, a 1 on first die and a 1 on second die. There are four ways that a sum of 5 can be rolled: 1-4, 2-3, 3-2, 4-1 (do not be confused here, 1-4 is a 1 on first die and a 4 on second and is distinct than a 4 on first and a 1 on second. When it helps, pretend that you are rolling one die and a friend is rolling other).

We wish our sample spaces to be equally likely if all possible.

Classical/Theoretical Probability:

When outcomes are equally likely, then probability of an event occurring is the number in event divided by the number in sample space.

P(E) = n(E)/n(S)

The probability of rolling a six on single roll of a die is 1/6 as there is only 1 way to roll six out of 6 ways it could be rolled.

The probability of obtaining a sum of 5 if rolling two dice is 4/36 = 1/9 because there are 4 ways to get a five and there are 36 ways to roll the dice (Fundamental Counting Principle - 6 ways to roll the first times 6 ways to roll the second).

Do not make the mistake of saying that the probability of rolling a sum of 5 is 1/11 as there is one 5 out of sample space of 11 sums (2 via 12). If the sample spaces are not equally likely, then don’t divide by the number in sample space.

Properties of Probabilities:

a) All the probabilities are between 0 and 1 inclusive.
b) The probability of 0 (zero) signifies an event is not possible, it can’t happen.
c) The probability of 1 signifies an event is certain to occur, it should happen.

Addition Rules:

When we wish to find out the probability of one event OR the other occurring, you add up their probabilities altogether.

This can lead to problems though, when they have something in common.

Probability of one or both of two events occurring is:

P(A or B) = P(A) + P(B) - P(A and B)

Mutually Exclusive Events:

The Mutually Exclusive Events encompass nothing in common. The intersection of two events is empty set. The probability of A and B both occurring is 0 (zero) as they cannot take place at similar time.

When two events are mutually exclusive, then the probability of one or other occurring is:

P(A or B) = P(A) + P(B)

Multiplication Rules:

When we wish to find out the probability of two events both occurring, then you require to apply the Fundamental Counting Principle. This principle can be expanded to probabilities.

Independent Events:

Independent Events are the events where one occurring does not modify the probability of the other occurring. Whenever events are independent, then the probability of both occurring is:

P(A and B) = P(A) * P(B)

We do not have time to get to probability very deeply. When we did, we would cover conditional probability: the probability of dependent events.

Complementary Events:

The root term in complementary is ‘complete’. Complementary events complete, or make entire. Complementary events are mutually exclusive, however when joined make the whole sample space.

The sign for the complement of event A is A'. Many books will place a bar over the set to point out its complement.

As complementary events are mutually exclusive, we can utilize the special addition rule to determine its probability. Moreover, complementary events are all inclusive; therefore they make the sample space whenever combined, therefore their probabilities have a sum of 1.

The sum of probabilities of complementary events is equal to 1.

P(A) + P(A') = 1
P(A') = 1 - P(A)

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