Download Lecture 7 Introduction to probability, Venn Diagram

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What can we say about probability?
◦ It is a measure of likelihood, uncertainty,
possibility, …
◦ And it is a number, numeric measure
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Set:
◦ A set is a collection of distinct objects, considered
as an object in its own right
◦ Example:
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A class of stat225 students
NBA teams
Result of a football game for a team (win, lose, tie)
Outcomes of rolling a die ( 1, 2, 3, 4, 5, 6)
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Experiment:
◦ A process that generates well-defined outcomes.
◦ *** You must know what could possibly happen
before performing the experiment.
◦ Example:
 Roll a die: YES
 ? Throw a stone out of window: NO.
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Sample space
◦ Don’t get confused with sample and sampling in
statistics.
◦ A sample space for an experiment is the set of ALL
experimental outcomes.
◦ Example:
 Toss a coin:
{Head, Tail}
 Take an exam or quiz:
{ All the possible grades }
 Inspection of a manufactured part: {defective, nondefective}
Some notes on sample space:
1. It could be finite or infinite, i.e., there
could be infinitely many possibilities, but you
still must know all of them.
Example, choosing a point on a segment,
there are infinitely many choices but you
know it must be on the segment.
2. Not all the sample points in the sample
space are equally likely.
Example: Getting a royal flush, full house
or just a pair in a poker game.
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Event:
◦ A collection of sample points.
 It is an outcome from an experiment that may include
one or more sample points.
◦ Examples:
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Toss a coin and get a head.
Roll a die and get a 5.
Toss a coin twice and get two heads.
Roll a die and get an even number.
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Complement of an event A:
◦ Notation: Ac
◦ Also an event
◦ Includes ALL sample points that are not in A.
 Example:
 A: Roll a die and get an even number,
 Ac : Roll a die and get an odd number.
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We use P(A) to represent the probability that
one event occurs.
Clearly: P(A) + P(Ac)=1.
A and Ac are called mutually exclusive, which
means any sample point call fall in either A
only or Ac only, but not both.
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Some of tossed a coin twice.
◦ 1. What is the sample space:
 { HH, HT, TH, TT }
◦ 2. Let A = { toss the coin twice and get two heads},
then P(A)=?
◦ 3. Let B = { toss the coin twice and get at least one
head}, then P(B)=?
◦ 4. What is Bc , and P(Bc )=?
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We have 20 fruits in a box, 10 apples, 6 pears
and 4 peaches.
◦ If A={pick a fruit from the box and get an apple},
then P(A)=?
◦ If B={pick a fruit from the box and it is NOT an
apple}, then P(B)=?
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50 balls are put in one box, 25 white, 15 red
and 10 green.
◦ A={ pick a ball and it is green }, P{A}=?
◦ B={pick a ball and it is colored}, P{B}=?
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Venn diagram:
◦ Illustrates the concept of complement.
◦ Somewhat like a crosstabulation.
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# 1. A class of 30 students took 2 midterms
during a semester. 22 of them passed the
first one and 26 of them passed the second
one. If 3 students failed both, find the
number of students who passed the each of
the two midterms and failed the other.
Answer: 21 passed both, 1 passed the first
but failed the second, 5 passed the second
but failed the first
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#2. We have a box and 30 balls, 16 are white
and the rest are colored; 9 are plastic and the
others are made of rubber. If there are 11
white rubber balls, show the breakdown of
balls by color and material.
Answer: 5 white plastic; 11 white rubber; 4
colored plastic and 10 colored rubber.
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#3. In a class of 30 students, 20 are male, 15
are white and 5 are black females. Assuming
only no students of other race in the class.
Find the break down of students by race and
gender.
Answer: 10 black male; 5 black female; 5
white female and 10 white male.
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In #1, what is the probability of finding a
student who passed at least one midterm?
◦ 27/30
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In #2, pick a ball at random, what is the
probability of getting a white plastic?
◦ 5/30
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In #3, pick a student from the class and what
is the probability of getting a white female?
◦ 5/30