 # Probability

Probability is a way of describing uncertainty in numerical terms. In this lesson, we review some of the terminology used in elementary probability theory: Probability of simple events, Rules of Probability, Mutually Exclusive and Independent Events.

A probability experiment, also called a random experiment, is an experiment for which the result, or outcome, is uncertain. We assume that all of the possible outcomes of an experiment are known before the experiment is performed, but which outcome will actually occur is unknown. The set of all possible outcomes of a random experiment is called the sample space, and any particular set of outcomes is called an event.

For example, consider a cube with faces numbered 1 to 6, called a 6-sided die. Rolling the die once is an experiment in which there are 6 possible outcomeseither 1, 2, 3, 4, 5, or 6 will appear on the top face. The sample space for this experiment is the set of numbers 1, 2, 3, 4, 5, and 6. Two examples of events for this experiment are
(i) rolling the number 4, which has only one outcome, and
(ii) rolling an odd number, which has three outcomes.

The probability of an event is a number from 0 to 1, inclusive, that indicates the likelihood that the event occurs when the experiment is performed. The greater the number, the more likely the event.

In general, for a random experiment with a finite number of possible outcomes, if each outcome is equally likely to occur, then the probability that an event A, written P(A), occurs is defined by the ratio Calculating the Probability of Simple Events.
This video shows the basic idea and a few simple examples of calculating the probability of simple events.

1. What is the probability of the next person you meeting having a phone number that ends in 5?
2. What is the probability of getting all heads if you flip three coins?
3. What is the probability that the person you meet next has a birthday in February) (Assume non-leap year)

This video introduces probability and determine the probability of basic events.

1. A bag contains 8 marbles numbered 1 to 8. a) What is the probability of selecting 2 from the bag?
b) What is the probability of selecting an odd number?
c) What is the probability of selecting a number greater than 6?
2. Using a standard deck of cards, determine each probability
a) P(face card)
b) P(5)
c) P(non face card)

## Rules of Probability

The following are general facts about probability.
 If an event E is certain to occur, then P E( ) = 1.
 If an event E is certain not to occur, then P E( ) = 0.
 If an event E is possible but not certain to occur, then 0 < P(E) <1.
 The probability that an event E will not occur is equal to 1 - P E( ).
 If E is an event, then the probability of E is the sum of the probabilities of the outcomes in E.
 The sum of the probabilities of all possible outcomes of an experiment is 1.

If E and F are two events of an experiment, we consider two other events related to E and F.
 The event that both E and F occur, that is, all outcomes in the set E ∩ F.
 The event that E or F, or both, occur, that is, all outcomes in the set E ∪ F.

This video explains the rules of probability.

## Mutually Exclusive and Independent Events

Events that cannot occur at the same time are said to be mutually exclusive.
For example, if a 6-sided die is rolled once, the event of rolling an odd number and the event of rolling an even number are mutually exclusive. But rolling a 4 and rolling an even number are not mutually exclusive, since 4 is an outcome that is common to both events.

For events E and F, we have the following rules.
 P(E or F) = P(E) + P(F) - P(E and F) which is the inclusion-exclusion principle applied to probability.
 If E and F are mutually exclusive, then P(E and F) = 0 and therefore, P(E or F) = P(E) + P(F)
 E and F are said to be independent if the occurrence of either event does not affect the occurrence of the other. If two events E and F are independent, then P(E and F) = P(E)P(F)

Probability - P(A ∪ B) and Mutually Exclusive Events.

Calculating Probability - " And " statements, independent events.
This video shows the basic idea, formula, and two examples.

How not to confuse the notions of mutually exclusive events and independent events
This video shows a way to get the definitions for mutually exclusive, and independent events right.
For two events, P(A) > 0 and P(B) > 0
• The events can't be both mutually exclusive and independent
• If A and B are independent then A and B are not mutually exclusive
• If A and B are mutually exclusive then A and B can't be independent.

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