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  Introduction to Probability, Classical Probability and Empirical Probability

Basic Definitions of Probability:

Probability Experiment: It is the process that leads to a well-defined result call outcomes.

Outcome: It is the result of a single trial of a probability experiment.

Sample Space: The probability experiment results a set of all possible outcomes termed as sample space.

Event: One or more outcomes of the probability experiment are termed as event.

Classical Probability: It uses the sample space to find out the numerical probability which an event will occur. It can also be termed as theoretical probability.

Equally Likely Events: Events that have similar probability of occurring.

Complement of an Event: Except the given events, the whole events in sample space are termed as complement of an event.

Empirical Probability: It determines the numerical probability by using a frequency distribution. The empirical probability is a relative frequency.

Subjective Probability: It uses the probability values based on an educated guess or approximation. It uses opinions and inexact information.

Mutually Exclusive Events: It states that two events that cannot occur at the same time.

Disjoint Events: It is the other name for the mutually exclusive events.

Independent Events: If the occurrence of one doesn’t affect the probability of other occurring then these are Independent events.

Two events are independent when the occurrence of one doesn’t affect the probability of other occurring.

Dependent Events: The two events are dependent when the first event influences the outcome or occurrence of the second event in a manner the probability is changed.

Conditional Probability: It is the probability of an event occurring given that the other event has already occurred.

Bayes' Theorem: It is a formula that permits one to find the probability that an event occurred as the outcome of a particular prior event.

Introduction to Probability:

Sample Spaces:

The set of all possible outcomes are termed as Sample Spaces. Though, some sample spaces are much better than others.

Let consider an experiment of flipping two coins. It is possible to acquire 0 heads, 1 head or 2 heads. Therefore, the sample space could be {0, 1, 2}. The other way to look at it is flip {HH, HT, TH, TT}. The second way is much better as each event is as equally likely to take place as any other.

It is extremely desirable to have events which are equally likely, whenever writing the sample space.

The other example is rolling two dice. The sums are {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Though, each of these is not equally likely. The only way to obtain a sum 2 is to roll a 1 on both dice; however you can get a sum of 4 by rolling a 1-3, 2-2, or 3-1. The following table describes a better sample space for the sum obtain whenever rolling two dice.

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Classical Probability:

By using a probability distribution, the table above lends itself to explain data in another way.

Let us consider the frequency distribution for the above sums.

Sum     Frequency     Relative Frequency
2          1                   1/36
3          2                   2/36
4          3                   3/36
5          4                   4/36
6          5                   5/36
7          6                   6/36
8          5                   5/36
9          4                   4/36
10        3                   3/36
11        2                   2/36
12        1                   1/36

We would have a probability distribution, if the first and last columns were written. The relative frequency of a frequency distribution is a probability of event occurring. If the events are equally likely then this holds true.

From the above we get the formula for classical probability. Probability of an event occurring is the number in event divided by the number in sample space. Again, this is holds true when the events are equally likely. Classical probability is the relative frequency of each and every event in the sample space if each event is equally likely.

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

Empirical Probability:

Empirical probability is mainly based on observation. The empirical probability of an event is the relative frequency of the frequency distribution based on observation.

P(E) = f/n

Probability Rules:

The probability rules are as shown below:

All probabilities are between 0 and 1:

0 <= P(E) <= 1

Sum of all the probabilities in sample space is 1:

There are various other rules that are as well important.

Probability of an event which can’t occur is 0:

The probability of any event that is not in the sample space is equal to zero.

Probability of an event which must occur is 1:

The probability of the sample space is equal to1.

Probability of an event not occurring is one minus the probability of its occurring.

P(E') = 1 - P(E)

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