Probability

Basic Concepts of Probability and Counting

Probability Experiment:… an action, or trial, through which

specific results are obtained.The result of a single trial is called an

OUTCOME.The set of all possible outcomes is

called the SAMPLE SPACE.An EVENT is a subset of the sample

space.

EX: Identify the sample space and determine the # of outcomes16. Guessing a student’s letter grade (A,

B, C, D, F) in a class.

18. Tossing three coins. (Hint… draw a tree diagram)

The Fundamental Counting PrincipleIf one event can occur in m ways and a

second event can occur in n ways, the number of ways the two events can occur in sequence is m · n

EX: For dinner you select one each from 3 appetizers, 4 entrees, and 2 desserts. How many different ‘meals’ can you make if you choose one from each category?

3 Types of Probability #1 Classical Probability(AKA Theoretical Probability): used

when each outcome in a sample space is equally likely to occur.

P(E) = probability of event E

P(E) = # of outcomes in event E Total # of outcomes in sample

space

#2 Empirical Probability(AKA Statistical Probability)Based on observations obtained from

probability experiments. Same as relative frequency of event.

P(E) = Frequency of Event E = f Total Frequency n

#3 Subjective ProbabilityResult from intuition, educated

guesses, and estimates.

As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical probability of the event.

The Law of Large Numbers

Classify as an example of classical, empirical, or subjective probability. The probability of choosing 6 numbers

from 1 to 40 that matches the 6 numbers drawn by a state lottery is 1/3,838,380 ≈ 0.00000026.

Rules of Probability0 < P(E) < 1 The probability of an event is between 0

and 1P(E) = 0 means the event CANNOT

occur.P(E) = 1 means the event is CERTAIN.ΣP(E) = 1The sum of the probabilities of all

outcomes in the sample space is one.

Complementary EventsThe complement of event E (denoted E’)

is the set of all outcomes in the sample space that are NOT part of event E.

P(E) + P(E’) = 1P(E’) = 1 – P(E)P(E) = 1 - P(E’)

Conditional Probability & the Multiplication Rule

Conditional Probability… the probability of an event occurring,

GIVEN that another event has occurred.

The conditional probability of event B occurring given that event A occurred is P(B | A)

Independent & Dependent EventsTwo events are INDEPENDENT if the occurrence of one does not affect the probability of the other event.

A and B are independent if…P(B | A) = P(B) or if…

P(A | B) = P(A)

Dependent or Independent?8. Returning a rented movie after the

due date and receiving a late fee.

12. A ball numbered 1 through 52 is selected from a bin, replaced, and a second numbered ball is selected from the bin.

The Multiplication RuleThe probability that A and B will

occur in sequence is:P(A and B) = P(A) · P(B | A)

If A and B are independent, use:P(A and B) = P(A) · P(B)