Download A chance process that leads to well

Math 150
Lecture Notes
Date
§4.1 Probability Basics
• Probability Experiment
A chance process that leads to well-defined results called
outcomes.
Roll a die
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• Sample Space
The set of all possible outcomes
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1,2,3,4,5,6
• Outcome
The result of a single trial of a probability experiment.
Getting a ’4’
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• Event
A set of outcome(s) of a probability experiment.
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Getting an even number
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J.M. Villalobos Probability types
Empirical Probability
◦
P (E) =
frequency of E
Total number of trials
Subjective Probability
Classical Probability
◦
Equally likely outcomes.
◦
P (E) =
n(E)
n(S)
.
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J.M. Villalobos Ex:
• Probability Experiment
A family decides to have three children.
• Sample Space
• Event
E = The family has two girls and one boy
A = The family has no boys
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J.M. Villalobos Ex:
• Probability Experiment
Nacho takes a test with three questions. Two questions are multiple choice (5 choices) and one True/False question.
• Sample Space
• Event
E = Nacho gets all three questions correct.
A = Nacho gets all three questions incorrect.
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J.M. Villalobos Probability rules
◦
0 ≤ P (E) ≤ 1
◦
P (E) = 0
◦
P (E) = 1
Complement of an Event
◦
P (E) = 1 − P (E)
◦
E = At least one
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J.M. Villalobos §4.2 Addition Rule
P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
Mutually Exclusive Events
Ex: A math class has 60 students. 35 students like chocolate, 25
students like strawberries and 15 students like both. Find the following
probabilities.
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J.M. Villalobos Ex: The following table summarizes the car manufacturer of 300
ECC-CC students.
Male
Female
American Japanese German
20
100
30
15
95
40
(a) What is the probability that a randomly selected student owns
an American Car?
(b) What is the probability that a randomly selected student is female or owns a German Car?
(c) What is the probability that a randomly selected student is Female or the student does NOT owns an American Car?
(d) What is the probability that the student owns a Japanese Car
or an American car?
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J.M. Villalobos §4.3 Independence
Two events A and B are independent (A⊥B) if the occurrence of A
does not affect the probability of B occurring.
If A⊥B then P (A ∩ B) = P (A)P (B)
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J.M. Villalobos Ex: Mickey Rats goes to dollar store and buys two alarms. The
probability that the alarms work any day is 90%. What is the probability that:
(a) both alarms will work tomorrow?
(b) he will wake up next Saturday?
Ex: According a recent survey 70% of Angelinos like chocolate. If
you randomly pick 6 Angelinos, what is the probability that at least
one of them likes chocolate?
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J.M. Villalobos Conditional Probability
The probability that event A will occur given that event B already
happened is
P (A|B) =
Ex:
Given
P (L) = 0.5,
P (A ∩ B)
P (B)
P (H) = 0.4,
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J.M. Villalobos P (L∩H) = 0.3 find
Ex: The following table summarizes the car make of 300 ECC-CC
students.
Male
Female
American Japanese German
20
100
30
15
95
40
(a) Are the events: American Car and Male Independent events?
(b) Given that a randomly chosen car is German what is the probability that is own by a woman?
(c) Given that a randomly chosen student is Male what is the probability that he owns Japanese Car?
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J.M. Villalobos Bayes Rule
Ex: Suppose that 46% of the population is male and that 82% own
a computer. If 71% of females own computers, what is the probability
that
(a) A randomly selected person does NOT own a computer?
(b) A randomly selected person owns a computer?
(c) If the person selected owns a computer, what is the probability
that the person was female?
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J.M. Villalobos Ex:
Suppose that 3% of the population has a rare disease. The
CDC has a test that is 98% effective if the person has the disease and
97% effective if the person does NOT have the disease. If a person is
a given this test what is the probability that
(a) the test will be positive?
(b) the test will be negative?
(c) If the test is positive, what is the probability that the person
actually has the the rare disease?
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J.M. Villalobos Counting
Fundamental Counting Principle
If event A can occur in m different ways and event B can occur in n
different ways then both events can occur in mn different ways.
Ex: SSN
Ex: Phone Numbers
Ex: Lottery
X X X X X X X X X
(562) X X X X X X X
X X X X X
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J.M. Villalobos X
Permutations
Combinations
Ex: This semester a math 150 class has 2 Republicans, 6 Democrats,
4 Independents, and 3 Communists. A 4-member committe will be
formed. What is the probability that:
(a) 1 Republican, 1 Democrat, and 2 Independent are chosen?
(b) 1 Republican and 3 Communist are chosen?
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