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Probability
Definitions
• Probability: The chance of an event occurring.
• Probability Experiments: A process that leads to welldefined results called outcomes.
• Examples of Experiments: Flipping a coin, Rolling a
die, Picking a card from a deck of cards.
• Outcomes: The result of a single trial of a probability
experiment.
• Sample Space: The set of the outcomes of a
probability experiment.
Definitions
• Events: Consist of one or more outcomes of a
probability experiment.
• 2 types of events
• Simple Event: An event with one outcome.
(Rolling a 3 on a die)
• Compound Event: An event with more than
one outcome. ( Rolling an even number on a
die)
Classical Probability
• Classical Probability: uses sample spaces to
determine the numerical probability that an
event will happen.
• Assumes that all outcomes in the sample space
are equally likely to occur.
•
number of outcomes in E
total number of outcomes in sample space
• Denoted by P(E).
Classical Examples
• What is the probability of rolling an even
number on a die?
• What is the probability of picking a king from
a deck of cards?
• What is the probability of rolling a number less
than 7 on a single die?
• All probabilities must fall within the following
range. ( 0 ≤ n ≤ 1 )
Complementary Events
• Complementary Events: The set of outcomes
in the sample space that are not included in the
outcomes of event E.
• Denoted by Ē (read “ E Bar” )
• Example 1: If we roll an even number on a die
its complement is rolling an odd.
• Example 2: Picking a red card from a deck its
complement is picking a black card.
• Rule: P (Ē) = 1 – P(E)
Empirical Probability
• Empirical Probability: Relies on actual
experiments to determine the likelihood of an
event. Using data from actual survey or other
type of research. Based on observation.
•
Frequency for the class
total frequencies in the distribution
• P (E) = f
n
Empirical Examples
• 20 fans from a Rutgers Football game were
asked how many home games they had been
to this season. The results are shown below.
Games attended
Frequency
2
5
3
4
4
7
5
4
Law of Large Numbers
• Law of Large Numbers: The more times that a
experiment is conducted the more likely the
empirical probability will approach the
theoretical probability.
• Example: Tossing a coin 50 or more times
should lead to the theoretical probability of
getting a tail = ½.
Subjective Probability
• Subjective probability: uses a probability value
based on an educated guess or estimate,
employing opinions and inexact information.
• Example 1: A doctor diagnosing that a patient
has a 30% chance of getting the flu this winter.
• Example 2: A weather forecaster stating that
there is a 60% chance of rain tomorrow.
Addition Rule for Probability
• Addition Rule: Determines the probability of a
compound event. An event that has 2 or more
outcomes.
• Example 1: Selecting a 6 or an 8 from a deck of
cards.
• Example 2: Getting a 1 or a 5 when rolling a die.
• Mutually exclusive events: Two events that can’t
occur at the same time.
• Not mutually exclusive: Two events that can
occur at the same time.
Exclusive or Not
• Determine if the following events are mutually
exclusive or not if a single card is chosen from
a deck.
1. Getting a 7 or getting a 9.
2. Getting a heart or a 10.
3. Getting a face card or a black card.
4. Getting a 2 or getting a face card.
Addition Rule 1
• When 2 events are mutually exclusive the
probability that A or B will occur is
P(A or B) = P(A) + P(B)
• Example 1: Probability of rolling a 3 or a 4 on
single die.
• Example 2: Probability of picking a queen or the
Ace of Spades when pulling a card from a deck
• Example 3: 7 cashiers, 2 managers, and 2 stock
boys work at a local store. What is the probability
if a person is selected at random that they are a
cashier or a stock boy?
Addition Rule 2
• When 2 events are not mutually exclusive the probability that A or B
will occur is:
P(A or B) = P(A) + P(B) – P(A and B)
• Example 1: Probability of selecting a 6 or a black card from a deck.
• Example 2: Probability of rolling a 2 or an even number on a single
die.
• Example 3: In a class there are 16 seniors and 11 juniors. 7 of the
seniors are male and 6 or the juniors are female. If a student is
selected at random what is the probability of selecting a senior or a
male?
• Example 4: The probability of a tourist visiting the Empire State
Building is 0.64 and of visiting Grand Central Station is 0.47. The
probability of visiting both on the same day is 0.33. Find the
probability that a tourist visits the Empire State Building or visits
Grand Central Station.
Multiplication Rule 1
• Used to find the probability of 2 or more
events that occur in sequence.
• Example: Rolling a die then flipping a coin
one can find the probability of rolling a 3 and
getting a tail.
• Independent Events: Events that do not affect
each other. Their probabilities of occurring are
also not affected.
• P(A and B) = P(A) ● P(B)
Rule 1 Examples
• Example 1: A spinner with 5 sections, numbered 1
-5 is spun and a card is picked from a deck. What
is the probability of the spinner landing on 3 and
picking a heart?
• Example 2: A coin is tossed and a card from a
deck is picked. What is the probability of landing
on heads and getting a face card?
• Example 3: Two dice are rolled, one at a time,
what’s the probability of rolling a 5 and another 5?
Multiplication Rule 2
• Used to find the probability of 2 or more events that occur
in sequence.
• Example: Probability of picking a king from a deck then
picking a 6 if not replacing the first card.
• Dependent Events: When the outcomes or occurrence of
the 1st event affects the 2nd event in such a way that the
probability is changed.
• Notation: P( B│A) This does not mean B divided by A.
It means that the probability event B occurs given that
event A has already happened. In example above P(B│A)
= 4/51
• Formula: P( A and B) = P (A) ● P(B│A)
Rule 2 Examples
• Example 1: What is the probability of picking an
ace and ace if the first card is not replaced?
• Example 2: 3 blue, 6 white, 2 red marbles all in a
bag. What is the probability of picking a blue,
white, and red marble if marbles are not replaced?
• Example 3: In a recent survey 49% of people said
they had renter’s insurance with Geico. Of those
22% said they also had car insurance with them.
If a person is selected at random, what’s the
probability that they have both?
Conditional Probability
• Conditional Probability: Imposes a condition
on the problem.
• The word “given” imposes the condition.
• The probability that the 2nd event B occurs
given the 1st event A has occurred can be found
by dividing the probability that both events
occurred by the probability that the 1st event
has occurred.
• Formula: P(B│A) = P (A and B) ÷ P (A)
Conditional Examples
• Example 1: At a country club 52% of the
members play bridge and swim, and 65%
swim. If a member is selected at random, find
the probability that the member plays bridge,
given that the member swims.
• Example 2: At a small college the probability a
students takes math and physics is 0.13. The
probability that a student takes math is 0.74.
Find the probability that the student is taking
physics given that he or she is taking math.
Conditional Examples
• Use the chart to answer the questions.
Employment Status
Owns a Card
Doesn't Own a Card
Employed
10
23
Unemployed
15
35
• If a person is selected at random find the probability the
person owns a credit card, given that the person is
employed.
• Find the probability the person is unemployed given that
the person does not own a credit card.
Tree Diagram Problems
• Example: A electronics company has 2
factories that produce televisions. Factory A
produces 60% and Factory B produces 40%.
8% of the televisions from Factory A are
defective while 15% from Factory B are
defective. If a tv is selected at random, find
the probability it is defective.
“At Least” Probabilities
• Can be tedious to find the result of each
possibility and then add them together.
• Easier to find the complement of the event and
then subtract that from 1.
• Example 1: Find the probability of getting at
least one King if you draw 4 cards from a deck. If
the card is replaced. Then if not replaced.
• Example 2: A coin is tossed 3 times. What is the
probability of at least one head?