Transcript STA 2023
STA 2023
Basic Concepts of Probability
and Counting
Probability Experiments
A probability
experiment is an action, or trial, through
which specific results are obtained.
The
result of a single trial is an outcome.
The
set of all possible outcomes of a probability
experiment is the sample space.
An
event is a subset of the sample space which may
have more than one outcome.
Example
1: Identify the sample space of the
probability experiment and determine the number of
outcomes in the sample space.
Tossing three Coins.
Start
H
T
H
T
H
T
H
T
H
T
H
T
H
T
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Sample
space:
{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
Number of Outcomes: 8
Simple Event
A simple
event is an event that consist of a single
outcome.
Example
2: For each event, determine the
number of outcomes and state whether the
event is simple or not.
Rolling an even number on a six-sided die.
3
outcomes, not a simple event
Pulling the Queen of Hearts out of a standard deck
of playing cards.
1outcome,
simple event.
The Fundamental
Counting Principle
Example
3: A shirt company has 4 designs each of
which can be made with short or long sleeves. There
are 7 color patterns available. How many different
shirts are available from this company?
56 shirts
Example
4: How many license plates can be made
consisting of 2 letters followed by 3 digits?
676,000 license plates
Types of Probability
Example
5: Identify the type of probability being used
The probability that a train will be in an accident on a
specific route is 1%
Empirical
The probability that interest rates will rise during the
summer is 0.05.
Subjective
The probability of winning California's Pick Three lottery
is 1/1000.
Classical
Other Properties
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
Range
of Probabilities
The probability of an event E is between 0 and 1,
inclusive.
Complements
The complement of event E is the set of all outcomes in
a sample space that are not included in event E.
Denoted as E’ (e prime)