Transcript Probability and Sample space

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 4.3
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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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, then A = {HT, TH}.
• Notation: The probability of an event A is denoted by P(A).
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Union and Intersection of events
• 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 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.
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Probability rules
1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.
2. If S is the sample space in a probability model, then P(S) = 1.
3. The complement of any event A is the event that A does not
occur, written as Ac . The complement rule states that
P(Ac) = 1 - P(A) .
4. 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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Venn diagram
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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 ; 0.01 ; 0.3 ; 0.6 ; 0.99 ; 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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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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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:
Color
Brown
Red
Yellow
Green
Orange
Blue
Probability
0.30
.20
.20
.10
.10
?
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?
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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.
(a) Find P(A) and P(B).
(b) Describe Ac in words and find P(Ac) by the complement rule.
(c) Describe {A or B} in words and find its probability by the
addition rule.
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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
countof outcom esin A countof outcom esin A
P A 

countof outcom esin S
k
• 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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General Addition rule for the unions of two 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).
• 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?
• 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?
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Conditional Probability
• The probability we assign to an event can change if we
know that some other event has occurred.
• When P(A) > 0, the conditional probability that B occurs
given the information that A occurs is
P A and B 
P(B| A) 
P  A
• 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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(a) 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?
(d) 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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Independent events
• 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.
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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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General Multiplication Rule
• The probability that both of two events A and B happen
together can be found by
P(A and B) = P(A)· P(B | A)
• Example 4.33 on page 317 in IPS.
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 = 0.1943.
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Bayes’s Rule
• If A and B are any events whose probabilities are not 0 or 1,
then
P(B| A)P( A)
P( A| B) 
P(B| A)P( A)  P(B| Ac)P( Ac)
• 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 any
outcome is in exactly one of these events. Then if C is any
other even whose probability is not 0 or 1,
P( Ai C ) 
P(C Ai ) P( Ai )
P(C A1 ) P( A1 )  P(C A2 ) P( A2 )      P(C Ak ) P( Ak )
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Exercise
(a)
(b)
(c)
(d)
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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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 0.01. 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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Exercise
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.
(a)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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Question 13 Term Test Summer 99
A space vehicle has 3 ‘o-rings’ which are located at various
field joint locations. Under current wheather conditions, the
probability of failure of an individual o-ring is 0.04.
(a) 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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Question 23 Final exam Dec 98
A large shipment of items is accepted by a quality checker
only if a random sample of 8 items contains no defective ones.
Suppose that in fact 5% of all items produced by this machine
are defective. Find the probability that the next two shipments
will both be rejected.
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Question 9 Final exam Dec 2001
You are going to travel Montreal, Ottawa, Halifax, and
Calgary, but the order is arbitrary. You put 4 marbles in a box,
each one labeled for one city, and draw randomly. The first
marble is the first city you will visit, the 2nd marble indicates
your 2nd stop etc.
What is the probability that you visit Ottawa just before or just
after you visit Montreal?
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