Transcript Section 11.1

Chapter 11
Probability
Copyright © 2015, 2011, and 2007 Pearson Education, Inc.
1
Chapter 11: Probability
11.1
11.2
11.3
11.4
11.5
Basic Concepts
Events Involving “Not” and “Or”
Conditional Probability and Events
Involving “And”
Binomial Probability
Expected Value and Simulation
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
2
Section 11-1
Basic Concepts
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
3
Basic Concepts
• Understand the basic terms in the language of
probability.
• Work simple problems involving theoretical and
empirical probability.
• Understand the law of large numbers (law of
averages).
• Find probabilities related to flower colors as
described by Mendel in his genetics research.
• Determine the odds in favor of an event and the
odds against an event.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
4
Probability
The study of probability is concerned with random
phenomena. Even though we cannot be certain
whether a given result will occur, we often can
obtain a good measure of its likelihood, or
probability.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
5
Probability
In the study of probability, any observation, or
measurement, of a random phenomenon is an
experiment. The possible results of the
experiment are called outcomes, and the set of all
possible outcomes is called the sample space.
Usually we are interested in some particular
collection of the possible outcomes. Any such
subset of the sample space is called an event.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
6
Example: Finding Probability When
Tossing a Coin
If a single fair coin is tossed, find the probability
that it will land heads up.
Solution
The sample space S = {h, t}, and the event whose
probability we seek is E = {h}.
P(heads) = P(E) = 1/2.
Since no coin flipping was actually involved, the
desired probability was obtained theoretically.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
7
Theoretical Probability Formula
If all outcomes in a sample space S are equally
likely, and E is an event within that sample space,
then the theoretical probability of the event E is
given by
number of favorable outcomes n( E )
P( E ) 

.
total number of outcomes
n( S )
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
8
Example: Flipping a Cup
A cup is flipped 100 times. It lands on its side 84
times, on its bottom 6 times, and on its top 10 times.
Find the probability that it will land on its top.
Solution
From the experiment it appears that
P(top) = 10/100 = 1/10.
This is an example of experimental, or empirical,
probability.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
9
Empirical Probability Formula
If E is an event that may happen when an experiment
is performed, then the empirical probability of
event E is given by
number of times event E occurred
P( E ) 
.
number of times the experiment was performed
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
10
Example: Finding Probability When
Dealing Cards
There are 2,598,960 possible hands in poker. If there
are 36 possible ways to have a straight flush, find
the probability of being dealt a straight flush.
Solution
36
P(straight flush) 
 .0000139
2,598,960
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
11
Example: Gender of a Student
A school has 820 male students and 835 female
students. If a student from the school is selected
at random, what is the probability that the student
would be a female?
Solution
number of female students
P(female) 
total number of students
835

 0.505
820 + 835
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
12
The Law of Large Numbers
As an experiment is repeated more and more
times, the proportion of outcomes favorable to
any particular event will tend to come closer and
closer to the theoretical probability of that event.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
13
Comparing Empirical and Theoretical
Probabilities
A series of repeated experiments provides an
empirical probability for an event, which, by
inductive reasoning, is an estimate of the event’s
theoretical probability. Increasing the number of
repetitions increases the reliability of the estimate.
Likewise, an established theoretical probability for
an event enables us, by deductive reasoning, to
predict the proportion of times the event will occur in
a series of repeated experiments. The prediction
should be more accurate for larger numbers of
repetitions.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
14
Probability in Genetics
Gregor Mendel, an Austrian monk, used the idea of
randomness to establish the study of genetics. To
study the flower color of certain pea plants, he found
that: Pure red crossed with pure white produces red.
Mendel theorized that red is “dominant”
(symbolized by R), while white is recessive
(symbolized by r). The pure red parent carried only
genes for red (R), and the pure white parent carried
only genes for white (r).
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
15
Probability in Genetics
Every offspring receives one gene from each parent
which leads to the tables below. Every second
generation is red because R is dominant.
1st to 2nd Generation
offspring
First
Parent
R
R
Second
Parent
r
r
Rr Rr
Rr Rr
2nd to 3rd Generation
offspring
First
Parent
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
R
r
Second
Parent
R
r
RR Rr
rR rr
16
Example: Probability of Flower Color
Referring to the 2nd to 3rd generation table (previous
slide), determine the probability that a third
generation will be
a) red
b) white
Base the probability on the sample space of equally
likely outcomes: S = {RR, Rr, rR, rr}.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
17
Example: Probability of Flower Color
Solution
S = {RR, Rr, rR, rr}.
a) Since red dominates white, any combination
with R will be red. Three out of four have an R,
so P(red) = 3/4.
b) Only one combination rr has no gene for red, so
P(white) = 1/4.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
18
Odds
Odds compare the number of favorable outcomes
to the number of unfavorable outcomes. Odds are
commonly quoted in horse racing, lotteries, and
most other gambling situations.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
19
Odds
If all outcomes in a sample space are equally
likely, a of them are favorable to the event E,
and the remaining b outcomes are unfavorable
to E, then the odds in favor of E are a to b, and
the odds against E are b to a.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
20
Example: Finding the Odds of Winning
a TV
200 tickets were sold for a drawing to win a new
television. If Matt purchased 10 of the tickets,
what are the odds in favor of Matt’s winning the
television?
Solution
Matt has 10 chances to win and 190 chances to
lose. The odds in favor of winning are 10 to 190,
or 1 to 19.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
21
Example: Converting from Probability
to Odds
Suppose the probability of rain today is 43%. Give
this information in terms of odds.
Solution
43
We can say that P (rain)  .43 
.
100
43 out of 100 outcomes are favorable, so 100 – 43 =
57 are unfavorable. The odds in favor of rain are 43
to 57 and the odds against rain are 57 to 43.
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
22
Example: Converting from Odds to
Probability
Your odds of completing a College Algebra class are
16 to 9. What is the probability that you will
complete the class?
Solution
There are 16 favorable outcomes and 9 unfavorable.
This gives 25 possible outcomes. So
16
P(completion) 
 0.64.
25
Copyright © 2016, 2012, and 2008 Pearson Education, Inc.
23