Probability IIntroduction to Probability ASatisfactory outcomes vs. total outcomes BBasic Properties CTerminology IICombinatory Probability AThe Addition Rule – “Or” 1.The special addition rule (mutually exclusive events) 2.The general addition rule (non-mutually exclusive events) BThe Multiplication Rule – “And” 1.The special multiplication rule (for independent events) 2.The general multiplication rule (for non-independent events)

Probability for Equally Likely Outcomes Suppose an experiment has N possible outcomes, all equally likely. Then the probability that a specified event occurs equals the number of ways, f, that the event can occur, divided by the total number of possible outcomes. In symbols Probability of a given event = Number of ways a given event can occur Total of all possible outcomes

Frequency distribution of annual income for U.S. families

Probability from Frequency Distributions What is the a priori probability of having an income between $15,000 and $24,999

Frequency distribution for students’ ages N = 40

Frequency distribution for students’ ages What is the likelihood of randomly selecting a student who is older than 20 but less than 22? What is the likelihood of selecting a student who’s age is an odd number? What is the likelihood of selecting a student who is either 21 or 23?

Sample space for rolling a die once

Possible outcomes for rolling a pair of dice

Probabilities of 2 throws of the die What is the probability of a 1 and a 3? What is the probability of two sixes? What is the probability of at least one 3? 2/36 1/36 12/36

The Sum of Two Die Tosses What is the probability that the sum will be 5? 7? What is the probability that the sum will be 10 or more? What is the probability that the sum will be either 3 or less or 11 or more? 4/36 6/36 3/36 + 3/36

Two computer simulations of tossing a balanced coin 100 times

Basic Properties of Probabilities Property 1: The probability of an event is always between 0 and 1, inclusive. Property 2: The probability of an event that cannot occur is 0. (An event that cannot occur is called an impossible event.) Property 3: The probability of an event that must occur is 1. (An event that must occur is called a certain event.)

A deck of playing cards

The event the king of hearts is selected 1/52

The event a king is selected 1/13 = 4/52

The event a heart is selected 1/4 = 13/52

The event a face card is selected 3/13=13/52

Sample Space and Events Sample space: The collection of all possible outcomes for an experiment. Event: A collection of outcomes for the experiment, that is, any subset of the sample space.

Probability Notation If E is an event, then P(E) stands for the probability that event E occurs. It is read “the probability of E”

Venn diagram for event E

Relationships Among Events (not E): The event that “E does not occur.” (A & B): The event that “both A and B occur.” (A or B): The event that “either A or B or both occur.”

Event (not E) where E is the probability of drawing a face card. 40/52=10/13

An event and its complement

The Complementation Rule For any event E, P(E) = 1 – P (~ E). In words, the probability that an event occurs equals 1 minus the probability that it does not occur.

Combinations of Events The Addition Rule – “Or” The special addition rule (mutually exclusive events) The general addition rule (non-mutually exclusive events) The Multiplication Rule – “And” The special multiplication rule (for independent events) The general multiplication rule (for non-independent events)

Venn diagrams for (a) event (not E ) (b) event (A & B) (c) event (A or B)

Event (B & C) 1/13 X 1/4 = 1/52

Event (B or C) 16/52 = 4/ /52-1/52

Event (C & D) 3/52 = 3/13 X 1/4

Mutually Exclusive Events Two or more events are said to be mutually exclusive if at most one of them can occur when the experiment is performed, that is, if no two of them have outcomes in common

Two mutually exclusive events

(a) Two mutually exclusive events (b) Two non-mutually exclusive events

(a) Three mutually exclusive events (b) Three non- mutually exclusive events (c) Three non-mutually exclusive events