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 outcomes
 16. 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
Principle
 If 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 Probability
 Result from intuition, educated guesses, and
estimates.
The Law of Large Numbers
 As an experiment is repeated over and over, the
empirical probability of an event approaches the
theoretical probability of the event.
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 Probability
0 < P(E) < 1
The probability of an event is between 0 and 1
P(E) = 0 means the event CANNOT occur.
P(E) = 1 means the event is CERTAIN.
ΣP(E) = 1
The sum of the probabilities of all outcomes in
the sample space is one.
Complementary Events
 The 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’) = 1
 P(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
Events
 Two 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 Rule
 The 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)