Download Chapter 14 sec 1

PROBABILITY
THEORY
Chapter 14 sec 1
Movie Quotes

"In this galaxy, there's a mathematical
probability of three million Earth type
planets. And in all of the universe
three million, million galaxies like
this. And in all of that, and perhaps
more, only one of each of us. Don't
destroy the one named 'Kirk.'"

Kirk: Mr. Spock, have you accounted for
the variable mass of whales and water in
your time re-entry program?
Spock: Mr. Scott cannot give me exact
figures, Admiral, so... I will make a guess.
Kirk: A guess? You, Spock? That's
extraordinary.
Famous quotes
Aristotle
The probable is what usually happens.
 Bertrand, Joseph
Calcul des probabilités
How dare we speak of the laws of
chance? Is not chance the antithesis of all
law?


Joseph Louis François Bertrand (March
11, 1822 – April 5, 1900, born and died
in Paris) was a French mathematician who
worked in the fields of number theory,
differential geometry, probability theory,
economics and thermodynamics.
Caesar, Julius
Iacta alea est. (The die is cast.)
 Doyle, Sir Arthur Conan
The Sign of Four
When you have eliminated the impossible,
what ever remains, however improbable,
must be the truth.

Random phenomena

What is random phenomena?
Occurrences
that vary from day to day
and case to case.
Weather conditions, rolling dice at craps or
Monopoly, drilling oil, driving your car.
We never know exactly how a random
phenomena will turn out, we often can
calculate a number called probability.
Experiment

Def.
Is
any observation of a random
phenomenon.
Outcome

Def.
The
different possible results of
the experiment
Sample Space

Def.
The
set of all possible outcomes for
an experiment.
Finding the sample space.
Example 1
We select an iPhone from a
production line and determine
whether it is defective.
 The sample space is;
{defective, nondefective}

Example 2
Three children are born to a family and
we note the birth order with respect to
gender.
 Make a tree diagram and find all the
possibilities.

{bbb,bbg,bgg,bgb,gbb,gbg,
ggb,ggg}
Event

Def.
In
probability theory, an event
is a subset of the sample space.
Write each event as a subset of the
sample space.
A tails occurs when we flip a single
coin.
 {Tails}
 Two girls and one boy are born in a
family.
 {ggb,gbg,bgg}

Probability of an outcome

Def.
In
a sample space is an number
between 0 and 1 inclusive. The
sum of the probabilities of all the
outcomes in the sample space must
be 1.
1.0
Certain
Likely to occur
0.5
50-50 Chance of occurring
Not likely to occur
0.0
Impossible
Probability of an event (E)

Def.
P(E) is defined as the sum of the
probabilities of the outcome that make
up E.


One way to determine probabilities
is to use empirical information.
Meaning we make observations and
assign probabilities based on those
observations.
Empirical assignment of Probabilities

If E is an event and we perform
an experiment several times, then
we estimate the probability of E
as follows;
Formula
𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠
𝑃 𝐸 =
𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑

The residents of a small town and the
surrounding area are divided over
the proposed construction of a spring
car racetrack in the town.
Table
Support
Oppose
In town
1,512
2,268
Surrounding
3,528
1,764
Problem

If a newspaper reporter randomly
selects a person to interview from
these people,
What is the probability that the
person supports the racetrack?
1512  3528
5
P( s) 

1512  3528  2268  1764 9

What are the odds in favor of the
person supporting the racetrack?
1512  3528
5

2268  1764
4

We normally say, 5 to 4
Cal. Probability when outcomes are
equally likely.

If E is an event in a sample space S with
all equally likely outcomes, then the
probability of E is given by the formula;
n( E )
P( E ) 
n( S )
Computing Probability of Events
What is the probability in a family
with three children that two of the
children are girls?
 Using example 2 that there are eight
outcomes in the sample set.
G={ggb,gbg,bgg}


Therefore, n(G) =3 and n(S) = 8.
n(G) 3
P(G) 

n( S ) 8
What is the probability that a total
of four shows when we roll two fair
dice?
 n(S) = what? What is the sample
space?
 The sample space for rolling two dice
has 36 ordered pairs of numbers.

Rolling a four, F = {(1,3),(2,2)(3,1)}
 Therefore,

n( F ) 3
1
P( F ) 


n( S ) 36 12
Basic Properties of Probability

Assume that S is a sample space for
some experiment and E is an event in
S.
1)
0  P( E )  1
P ( )  0
3) P ( S )  1
2)
Probability formula for computing odds

If E’ is the complement of the event E,
then the odds against E are
P( E ' )
P( E )
Example problem
Suppose that the probability of
the Saints winning the Super
Bowl is 0.15. What are the
odds against the Saints winning
the Super Bowl.

1-0.15 = 0.85
 This answer is the complement, P(E’)
 0.15 is the probability of E, P(E)
 The odds against the Saints are

P( E ' ) 0.85 0.85 x100 85 17




P( E ) 0.15 0.15 x100 15 3
Therefore, we would say
that the odds against the
Saints winning the Super
Bowl are 17 to 3.
