1. Probability and Sample Space
We call a phenomenon random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions.
The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, probability is a long-term relative frequency.
Example: Tossing a coin: P(H) = ?
The sample space of a random phenomenon is the set of all possible outcomes.
Example: Toss a coin the sample space is S = {H, T}.
Example: From rolling a die, S = {1, 2, 3, 4, 5, 6}.
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2. Events
An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.
Example:
Take the sample space (S) for two tosses of a coin to be the 4 outcomes {HH, HT, TH TT}.
Then exactly one head is an event, call it A, and A = {HT, TH}.
Notation: The probability of an event A is denoted by P(A).
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3. Union and Intersection of Events
The intersection of any collection of events is the event that all of the events occur.
Example: The event {A and B} is the intersection of A and B, it is the event that both A and B occur.
The union of any collection of events is the event that at least one of the events in the collection occurs.
Example: The event {A or B} is the union of A and B, it is the event that at least one of A or B occurs (either A occurs or B occurs or both occur).
The compliment of an event A is the set of all elements in S that are not in A.
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4. Probability Rules
The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.
If S is the sample space in a probability model, then P(S) = 1.
The complement rule states that
P(A’) = 1 - P(A) .
Two events A and B are disjoint if they have no outcomes in common and so can never occur together.
If A and B are disjoint then
P(A or B) = P(A U B) = P(A) + P(B) .
This is the addition rule for disjoint events and can be
extended for more than two events
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5. Venn Diagram
Events, relationship between events and their probabilities are best described with vann diagrams.
Example….
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6. Question
Probability is a measure of how likely an event is to occur. Match
one of the probabilities that follow with each statement about an
event. (The probability is usually a much more exact measure of
likelihood than is the verbal statement.)
0 ; ; 0.3 ; 0.6 ; ; 1
(a) This event is impossible. It can never occur.
(b) This event is certain. It will occur on every trial of the
random phenomenon.
(c) This event is very unlikely, but it will occur once in a
while in a long sequence of trials.
(d) This event will occur more often than not.
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7. Probabilities for Finite Number of Outcomes
The individual outcomes of a random phenomenon are always disjoint. So the addition rule provides a way to assign probabilities to events with more then one outcome.
Assign a probability to each individual outcome. These probabilities must be a number between 0 and 1 and must have sum 1.
The probability of any event is the sum of the probabilities of the outcomes making up the event.
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8. Question
If you draw an M&M candy at random from a bag of the candies, the candy
you draw will have one of six colors. The probability of drawing each color
depends on the proportion of each color among all candies made.
(a) The table below gives the probability of each color for a randomly chosen plain M&M:
What must be the probability of drawing a blue candy?
(b) What is the probability that a plain M&M is any of red, yellow, or orange?
(c) What is the probability that a plain M&M is not red?
Color 
Brown
Red
Yellow
Green
Orange
Blue
Probability
0.30
.20
.10 ?
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9. Question Choose an American farm at random and measure its size in
acres. Here are the probabilities that the farm chosen falls in
several acreage categories:
Let A be the event that the farm is less than 50 acres in size, and
let B be the event that it is 500 acres or more.
Find P(A) and P(B).
(b) Describe Ac in words and find P(Ac) by the complement rule.
Describe {A or B} in words and find its probability by the
addition rule.
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10. Equally Likely Outcomes
If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is
Example:
A pair of fair dice are rolled. What is the probability that the
2nd die lands on a higher value than does the 1st ?
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11. General Addition Rule for the Unions of Events
If events A and B are not disjoint, they can occur together.
For any two events A and B
P(A or B) = P(A U B) = P(A) + P(B) - P(A and B).
This rule can be extended to more then two events.
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12. Exercise
A retail establishment accepts either the American Express or the
VISA credit card. A total of 24% of its customers carry an American
Express card, 61% carry a VISA card, and 11% carry both. What
percentage of its customers carry a card that the establishment will
accept?
Solution….
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13. Exercise
Among 33 students in a class, 17 earned A’s on the midterm exam,
14 earned A’s on the final exam, and 11 did not earn A’s on either
examination. What is the probability that a randomly selected
student from this class earned A’s on both exams?
Solution….
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14. Conditional Probability
The probability we assign to an event can change if we know that some other event has occurred.
Example…
When P(A) > 0, the conditional probability that B occurs given the information that A occurs is
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15. Example Here is a two way table of all suicides committed in a recent
year by sex of the victim and method used.
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16. What is the probability that a randomly selected suicide victim
is male?
(b) What is the probability that the suicide victim used a firearm?
(c) What is the conditional probability that a suicide used a
firearm, given that it was a man?
Given that it was a woman?
Describe in simple language (don't use the word “probability”)
what your results in (c) tell you about the difference between
men and women with respect to suicide.
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17. Independent Events P(B | A) = P(B) . P(A and B) = P(A)·P(B) .
Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. That is, if A and B are independent then,
P(B | A) = P(B) .
Multiplication rule for independent events
If A and B are independent events then,
P(A and B) = P(A)·P(B) .
The multiplication rule applies only to independent events; we can
not use it if events are not independent.
This rule can be extended to more then two events.
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18. Example
The gene for albinism in humans is recessive. That is, carriers of this
gene have probability 1/2 of passing it to a child, and the child is
albino only if both parents pass the albinism gene. Parents pass their
genes independently of each other. If both parents carry the albinism
gene, what is the probability that their first child is albino?
If they have two children (who inherit independently of each other),
what is the probability that
(a) both are albino?
(b) neither is albino?
(c) exactly one of the two children is albino?
If they have three children (who inherit independently of each other),
what is the probability that at least one of them is albino?
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19. General Multiplication Rule
The probability that both events A and B happen together can be
found by
P(A and B) = P(A)· P(B | A)
Example
29% of Internet users download music files and 67% of the
downloaders say they don’t care if the music is copyrighted. The
percent of Internet users who download music (event A) and don’t
care about copyright (event B) is
P(A and B) = P(A)· P(B | A) = 0.29·0.67 =
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20. Theorem of Total Probability
Definition:
A partition of the sample space S is a countable collection of events such that
Theorem:
If is a partition of S such that then
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21. Bayes’s Rule
If A and B are any events whose probabilities are not 0 or 1, then
Example: Following exercise using tree diagram.
Suppose that A1, A2,…, Ak are disjoint events whose probabilities are not 0 and add to exactly 1. That is, A1, A2,…, Ak is a partition of the sample space. Then if C is any other even whose probability is not 0 or 1, then
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22. Exercise
The fraction of people in a population who have a certain disease is
0.01. A diagnostic test is available to test for the disease. But for a
healthy person the chance of being falsely diagnosed as having the
disease is 0.05, while for someone with the disease the chance of
being falsely diagnosed as healthy is 0.2. Suppose the test is
performed on a person selected at random from the population.
What is the probability that the test shows a positive result?
What is the probability that a person selected at random is one who has the disease but was diagnosed healthy?
What is the probability that the person is correctly diagnosed and is healthy?
If the test shows a positive result, what is the probability this person actually has the disease?
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23. Exercise An automobile insurance company classifies drivers as class A
(good risks), class B (medium risks), and class C (poor risks). Class
A risks constitute 30% of the drivers who apply for insurance, and
the probability that such a driver will have one or more accidents in
any 12-month period is The corresponding figures for class B
are 50% and 0.03, while those for class C are 20% and 0.10.
The company sells Mr. Jones an insurance policy, and within 12
months he had an accident.
What is the probability that he is a class A risk?
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24. Exercise approximately as follows: 37% type A, 13% type B, 44% type
The distribution of blood types among white Americans is
approximately as follows: 37% type A, 13% type B, 44% type
O, and 6% type AB. Suppose that the blood types of married
couples are independent and that both the husband and wife
follow this distribution.
An individual with type B blood can safely receive transfusions only
from persons with type B or type O blood. What is the probability
that the husband of a woman with type B blood is an acceptable
blood donor for her?
(b) What is the probability that in a randomly chosen couple the wife has type B blood and the husband has type A?
(c) What is the probability that one of a randomly chosen couple has type A blood and the other has type B?
(d) What is the probability that at least one of a randomly chosen couple has type O blood?
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25. Question from Past Term Test
A space vehicle has 3 ‘o-rings’ which are located at various field joint locations. Under current whether conditions, the probability of failure of an individual o-ring is 0.04.
A disaster occurs if any of the o-rings should fail. Find the
probability of a disaster. State any assumptions you are making.
(b) Find the probability that exactly one o-ring will fail.
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26. Random Variables Example:
We roll a fair coin 4 times. Suppose we are interested in the number of heads in the 4 rolls. Let X = number of 5’s. Then X could be 0, 1, 2, 3, 4.
X = 0 corresponds to the 1 elements of our 16 elements of the sample space S.
X = 1 corresponds to the 4 elements etc.
X is an example of a random variable.
A random variable is a function that maps a real number to each element in the sample space.
Probability models are often stated in terms of random variables.
E.g. - model for the # of H’s in 10 flips of a coin.
- model for the height of a randomly chosen person.
- model for size of a queue.
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27. Discrete Random Variable
Definition:
A random variable X is said to be discrete if it can take only a finite or countable infinite number of distinct values.
A discrete random variable X maps the sample space S onto a countable set. Define a probability mass function (pmf) or frequency function on X such that
Where the sum is taken over all possible values of X.
The probability distribution of a discrete random variable X is represented by a formula, a table or a graph which provides the list of all possible values that X can take and the pmf for each value.
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28. Examples of Discrete Random Variables
1. Consider the experiment of tossing a coin. Define a random variable as follows: X = 1 if a H comes up
= 0 if a T comes up.
This is an example of a Bernoulli r.v. The Probability function of
X is given in the following table X
P(X = x) 1 p
1- p
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29. The probability function of X is given in the following table.
Let X be a r.v counting the number of girls in a family with 3 children.
The probability function of X is given in the following table.
Toss a coin 4 times. Let X be the number of H’s. Find the
probability function of X. Draw a probability histogram.
Toss a coin until the 1st H. Let X be the number of T’s before the
1st H. Find the probability function of X. x
P(X = x) 1 2 3
(0.5)3 = 0.125
3·(0.5)3 = 0.375
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30. Distribution Function of Random Variables
A cumulative distribution function (cdf) of a random variable X is a function defined by
If X is a discrete random variable with pmf for x = 0, 1, 2, … then
where is the greatest integer ≤ v.
Example…
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31. Properties of Distribution Function
F is monotone, non decreasing i.e. F(x) ≤ F(y) if x ≤ y.
As x  - ∞ , F(x)  0
As x  ∞ , F(x)  1
F(x) is continuous from the right
For a < b
Why?
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32. Continuous Random Variable - Introduction
If the sample space contains an infinite number of possible outcomes equal to the number of points of a line segment, it is called a continuous sample space.
A continuous random variable X takes all values in an interval of numbers.
The probability distribution of X is often described by a density curve. It is the plot of the probability density function.
The total area under a density curve is 1.
The probability of any event is the area under the density curve and above the value of X that make up the event.
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33. Example
The density function of a continuous r. v. X is given in the graph below.
Find
1) P(X < 7)
2) P(6 < X < 8)
3) P(X = 7)
4) P(5.5 < X < 7 or 8 < X < 9)
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34. Formal Definition of Continuous Random Variable
A random variable X is continuous if its cumulative distribution function can be written in the form
for some non-negative function f.
fX(x) is the (Probability) Density Function of X.
The Density Function fX(x) of X is simply the derivative of the cdf (if it exists).
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35. Facts and Properties of Pdf
Properties of Probability Density Function (pdf)
Any function satisfying these two properties is a probability density function (pdf) for some random variable X.
Note: fX (x) does not give a probability.
For continuous random variable X with density f
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36. Example
Kerosene tank holds 200 gallons; The model for X the weekly demand is given by the following density function
Check if this is a valid pdf.
Find the cdf of X.
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