Stats: Probability

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Definitions

Process which leads to well-defined results call outcomes

Outcome

The result of a single trial of a probability experiment

Sample Space

Set of all possible outcomes of a probability experiment

Event

One or more outcomes of a probability experiment

Classical Probability

Uses the sample space to determine the numerical probability that an event will happen. Also called theoretical probability.

Equally Likely Events

Events which have the same probability of occurring.

Complement of an Event

All the events in the sample space except the given events.

Uses a frequency distribution to determine the numerical probability. An empirical probability is a relative frequency.

Subjective Probability

Uses probability values based on an educated guess or estimate. It employs opinions and inexact information.

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Mutually Exclusive Events

Two events which cannot happen at the same time.

Disjoint Events

Another name for mutually exclusive events.

Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other occurring.

Dependent Events

Two events are dependent if the first event affects the outcome or occurrence of the second event in a way the probability is changed.

Conditional Probability

The probability of an event occurring given that another event has already occurred.

Bayes’ Theorem

A formula which allows one to find the probability that an event occurred as the result of a particular previous event.

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Table of Contents

Stats: Introduction to Probability

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Sample Spaces

A sample space is the set of all possible outcomes. However, some sample spaces are better than others.

Consider the experiment of flipping two coins. It is possible to get 0 heads, 1 head, or 2 heads. Thus, the sample space could be {0, 1, 2}. Another way to look at it is flip { HH, HT, TH, TT }. The second way is better because each event is as equally likely to occur as any other.

When writing the sample space, it is highly desirable to have events which are equally likely.

Another example is rolling two dice. The sums are { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }. However, each of these aren’t equally likely. The only way to get a sum 2 is to roll a 1 on both dice, but you can get a sum of 4 by rolling a 1-3, 2-2, or 3-1. The following table illustrates a better sample space for the sum obtain when rolling two dice.

### Classical Probability

The above table lends itself to describing data another way — using a probability distribution. Let’s consider the frequency distribution for the above sums.

If just the first and last columns were written, we would have a probability distribution. The relative frequency of a frequency distribution is the probability of the event occurring. This is only true, however, if the events are equally likely.

This gives us the formula for classical probability. The probability of an event occurring is the number in the event divided by the number in the sample space. Again, this is only true when the events are equally likely. A classical probability is the relative frequency of each event in the sample space when each event is equally likely.

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

Empirical Probability

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

P(E) = f / n

Probability Rules

There are two rules which are very important.

All probabilities are between 0 and 1 inclusive

0