Transcript 3.1 Basic Concepts of Probability

Basic Concepts of Probability
Objectives/Assignment
• How to identify the sample space of a
probability experiment and to identify simple
events.
• How to distinguish between classical
probability, empirical probability and
subjective probability.
• How identify and use properties of probability.
Definitions
• Probability experiment: An action through which specific results (counts,
measurements, or responses) are obtained.
• Outcome: The result of a single trial in a probability experiment.
• Sample Space: The set of all possible outcomes of a probability
experiment
• Event: One or more outcomes and is a subset of the sample space
HINTS TO REMEMBER:
• Probability experiment: Roll a six-sided die
• Sample space: {1, 2, 3, 4, 5}
• Event: Roll an even number (2, 4, 6)
• Outcome: Roll a 2, {2}
Identifying Sample Space of a Probability
Experiment
• A probability experiment consists of tossing a
coin and then rolling a six-sided die. Describe the
sample space.
• There are two possible outcomes when tossing a
coin—heads or tails. For each of these there are
six possible outcomes when rolling a die: 1, 2, 3,
4, 5, and 6. One way to list outcomes for actions
occurring in a sequence is to use a tree diagram.
From this, you can see the sample space has 12
outcomes.
Tree Diagram for Coin and Die Experiment
• {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4,
H T5, T6}
• |
1
2
3
4
H1
H2
H3
H4
1
5
H5
6
1
2
3
4
5
H6
T1
T2
T3
T4
T5
T
6
T6
Some more
• A probability experiment consists of
recording a response to the survey
statement below and tossing a coin.
Identify the sample space.
• Survey: There should be a limit to the
number of terms a US senator can serve.
•
Agree
Disagree
No opinion
Some more
• A probability experiment consists of recording
a response to the survey statement below and
tossing a coin. Identify the sample space.
• A. Start a tree diagram by forming a branch
for each possible response to the survey.
Agree
Disagree
No opinion
Some more
• A probability experiment consists of recording
a response to the survey statement below and
tossing a coin. Identify the sample space.
• B. At the end of each survey response branch,
draw a new branch for each possible coin
outcome.
No opinion
Agree
Disagree
H
T
H
T
H
T
Some more
• A probability experiment consists of recording
a response to the survey statement below and
tossing a coin. Identify the sample space.
• C. Find the number of outcomes in the
sample space. In this case -- 6
Agree
H
T
Disagree
H
T
No opinion
H
T
Some more
• A probability experiment consists of recording
a response to the survey statement below and
tossing a coin. Identify the sample space.
• D. List the sample space
• {Ah, At, Dh, Dt, Nh, Nt}
Agree
H
T
Disagree
H
T
No opinion
H
T
Simple events
• Simple Event: An event that consists of a single outcome.
• Decide whether the event is simple or not. Explain your reasoning:
• 1. For quality control, you randomly select a computer chip from a batch
that has been manufactured that day. Event A is selecting a specific
defective chip. (Simple because it has only one outcome: choosing a
specific defective chip. So, the event is a simple event.
• 2. You roll a six-sided die. Even B is rolling at least a 4.
• B has 3 outcomes: rolling a 4, 5 or 6. Because the event has more than
one outcome, it is not simple.
Simple events
• You ask for a student’s age at his or her last
birthday. Decide whether each even is simple or
not:
• 1. Event C: The student’s age is between 18 and
23, inclusive.
• A. Decide how many outcomes are in the event.
• B. State whether the event is simple or not.
Simple events
• You ask for a student’s age at his or her last birthday. Decide whether
each even is simple or not:
• 1. Event C: The student’s age is between 18 and 23, inclusive.
• A. Decide how many outcomes are in the event. The student’s age can
be {18, 19, 20, 21, 22, or 23} = 6 outcomes
• B. State whether the event is simple or not. Because there are 6
outcomes, it is not a simple event.
Simple events
• You ask for a student’s age at his or her last
birthday. Decide whether each even is simple or
not:
• 1. Event D: The student’s age is 20.
• A. Decide how many outcomes are in the event.
• B. State whether the event is simple or not.
Simple events
• You ask for a student’s age at his or her last birthday.
Decide whether each even is simple or not:
• 1. Event D: The student’s age is 20.
• A. Decide how many outcomes are in the event. There
is only 1 outcome – the student is 20.
• B. State whether the event is simple or not. Since
there is only one outcome, it is a simple event.
Everything beyond this is optional…
• If you want to continue to learn about
probability, read on! If not, no big deal! 
Types of probability
• Three types of probability:
• 1. Classical probability
• 2. empirical probability
• 3. subjective probability
Classical probability
• P(E) =
# of outcomes in E___________
•
Total # of outcomes in sample space
• You roll a six-sided die. Find the probability of
the following:
• 1. Event A: rolling a 3
• 2. Event B: rolling a 7
• 3. Event C: rolling a number less than 5.
Classical probability
• P(E) = # of outcomes in E___________
•
Total # of outcomes in sample space
• You roll a six-sided die. Find the probability of
the following:
• First when rolling a six-sided die, the sample
space consists of six outcomes {1, 2, 3, 4, 5, 6}
Classical probability
• P(E) =
# of outcomes in E___________
•
Total # of outcomes in sample space
• You roll a six-sided die. Find the probability of
the following:
• 1. Event A: rolling a 3 There is one outcome in
event A = {3}. So,
•
P(3) = 1/6 = 0.167
Classical probability
• P(E) =
# of outcomes in E___________
•
Total # of outcomes in sample space
• You roll a six-sided die. Find the probability of
the following:
• 2. Event B: rolling a 7 Because 7 is not in the
sample space, there are no outcomes in event B.
So,
• P(7) = 0/6 = 0
Classical probability
• P(E) =
# of outcomes in E___________
•
Total # of outcomes in sample space
• You roll a six-sided die. Find the probability of
the following:
• 3. Event C: rolling a number less than 5. There
are four outcomes in event C {1, 2, 3, 4}. So
• P(number less than 5) = 4/6 = 2/3 ≈0.667
Example
• You select a card from a standard deck. Find the
probability of the following:
• 1. Event D: Selecting a seven of diamonds.
• 2. Event E: Selecting a diamond
• 3. Event F: Selecting a diamond, heart, club or
spade.
• A. Identify the total number of outcomes in the
sample space.
• B. Find the number of outcomes in the event.
• C. Use the classical probability formula.
Example
• You select a card from a standard deck. Find the
probability of the following:
• 1. Event D: Selecting a seven of diamonds.
• A. Identify the total number of outcomes in the
sample space. (52)
• B. Find the number of outcomes in the event. (1)
• C. Use the classical probability formula.
• P(52) = 1/52 or 0.0192
Example
• You select a card from a standard deck. Find the
probability of the following:
• 2. Event E: Selecting a diamond
P(52)number
= 1/52 orof0.0192
• A. Identify the total
outcomes in the
sample space. (52)
• B. Find the number of outcomes in the event. (13)
• C. Use the classical probability formula.
• P(52) = 13/52 or 0.25
Example
• You select a card from a standard deck. Find the
probability of the following:
• 3. Event F: Selecting a diamond, heart, club or spade.
• A. Identify the total number of outcomes in the
sample space. (52)
• B. Find the number of outcomes in the event. (52)
• C. Use the classical probability formula.
• P(52) = 52/52 or 1.0
Empirical Probability
• Based on observation obtained from
probability experiments. The empirical
probability of an event E is the relative
frequency of event E
FrequencyofeventE f
P( E ) 

Totalfrequency
n
Example
• A pond containing 3 types of fish: bluegills, redgills, and
crappies. Each fish in the pond is equally likely to be caught.
You catch 40 fish and record the type. Each time, you release
the fish back in to the pond. The following frequency
distribution shows your results.
Fish Type
# of Times Caught, (f)
Bluegill
13
Redgill
17
Crappy
10
f = 40
If you catch another fish, what is the probability that it
is a bluegill?
Empirical Probability ex. continued
• The event is “catching a bluegill.” In your
experiment, the frequency of this event is 13.
because the total of the frequencies is 40, the
empirical probability of catching a bluegill is:
• P(bluegill) = 13/40 or 0.325
Example
• An insurance company determines that in every
100 claims, 4 are fraudulent. What is the
probability that the next claim the company
processes is fraudulent.
• A. Identify the event. Find the frequency of the
event. (finding the fraudulent claims, 4)
• B. Find the total frequency for the experiment.
(100)
• C. Find the relative frequency of the event.
•
P(fraudulent claim) = 4/100 or .04
Law of large numbers
• As you increase the number of times a
probability experiment is repeated, the
empirical probability (relative frequency) of an
event approaches the theoretical probability
of the event. This is known as the law of large
numbers.
Using frequency distributions to find Probability
• You survey a sample of 1000 employees at a
company and record the ages of each. The results
are shown below. If you randomly select another
employee, what is the probability that the
employee is between 25 and 34 years old?
Employee Ages
Frequency
Age 15-24
54
Age 25-34
366
Age 35-44
233
Age 45-54
180
Age 55-64
125
65 and over
42
f = 1000
Using frequency distributions to find Probability
• P(age 25-34) = 366/1000 = 0.366
Employee Ages
Frequency
Age 15-24
54
Age 25-34
366
Age 35-44
233
Age 45-54
180
Age 55-64
125
65 and over
42
f = 1000
Example
• Find the probability that an employee chosen
at random is between 15 and 24 years old.
• P(age 15-24) = 54/1000 = .054
Employee Ages
Frequency
Age 15-24
54
Age 25-34
366
Age 35-44
233
Age 45-54
180
Age 55-64
125
65 and over
42
f = 1000
Subjective Probability
• Subjective probability result from intuition,
educated guesses, and estimates. For instance,
given a patient’s health and extent of injuries,
a doctor may feel a patient has 90% chance of a
full recovery. A business analyst may predict
that the chance of the employees of a certain
company going on strike is .25
• A probability cannot be negative or greater
than 1, So, the probability of and event E is
between 0 and 1, inclusive. That is 0  P(E)  1
Classifying types of Probability
• Classify each statement as an example of classical
probability, empirical probability or subjective
probability. Explain your reasoning.
• 1. The probability of your phone ringing during
the dinner hour is 0.5
• This probability is most likely based on an
educated guess. It is an example of subjective
probability.
Classifying types of Probability
• Classify each statement as an example of classical
probability, empirical probability or subjective
probability. Explain your reasoning.
• 2. The probability that a voter chosen at random
will vote Republican is 0.45.
• This statement is most likely based on a survey of
voters, so it is an example of empirical
probability.
Classifying types of Probability
• Classify each statement as an example of classical
probability, empirical probability or subjective
probability. Explain your reasoning.
• 3. The probability of winning a 1000-ticket raffle
with one ticket is 1/1000.
• Because you know the number of outcomes and
each is equally likely, this is an example of
classical probability.
Classifying types of Probability
• Classify each statement as an example of classical
probability, empirical probability or subjective
probability. Explain your reasoning.
• Based on previous counts, the probability of a
salmon successfully passing through a dam on
the Columbia River is 0.85.
• Event: = salmon successfully passing through a
dam on the Columbia River.
• Experimentation, Empirical probability.
Properties of Probability
• The sum of the probabilities of all outcomes in a
sample space is 1 or 100%. An important result
of this fact is that if you know the probability of
event E, you can find the probability of the
complement of event E.
• The complement of Event E, is the set of all
outcomes in a sample space that are not included
in event E. The complement of event E is
denoted by E’ and is read as “E prime.”
Example
• For instance, you roll a die and let E be the event “the number is at least
5,” then the complement of E is the event “the number is less than 5.” In
other words, E = {5, 6} and E’ = {1, 2, 3, 4}
• Using the definition of the complement of an event and the fact that the
sum of the probabilities of all outcomes is 1, you can determine the
following formulas:
• P(E) + P(E’) = 1
• P(E) = 1 – P(E’)
• P(E’) = 1 – P(E)
Finding the probability of the complement of an
event
• Use the frequency distribution given in example 5
to find the probability of randomly choosing an
employee who is not between 25 and 34.
• P(age 25-34) = 366/1000 = 0.366
• So the probability that an employee is not
between the ages of 25-34 is
• P(age is not 25-34) = 1 – 366/1000 = 634/1000 =
0.634
Example: Use the frequency distribution in Example 4 to find the probability
that a fish that is caught is NOT a redgill.
• A. Find the probability that the fish is a redgill.
• 17/40 = .425
• B. Subtract the resulting probability from 1—
• 1-.425 = .575
• C. State the probability as a fraction and a decimal.
• 23/40 = .575
Fish Type
# of Times Caught, (f)
Bluegill
13
Redgill
17
Crappy
10
f = 40