Download Probability - Seattle Central College

Chapter 3:
Probability
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Section 3-1: Introduction

The decision to work out or foreclose a
loan depends on the probability of success
or failure of the workout. This will
determine how much money we will
recover from the loan.
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Definitions




A Trial is an activity where the result is
unknown. (this is sometimes called an
experiment, or a random experiment)
An Outcome is one specific result of a trial.
A Sample Space is the set of all possible
outcomes. It is usually represented by a capital
S, but we will use the symbol § to represent
sample space. (S is needed for something else)
A Probability is the proportion of times a
particular event will occur. It is usually
represented by a capital P.
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Flip a coin 10 times.
 Record the number of times a “head”
comes up.
 Add up the total number for the whole
class
 What do you observe?
 How many heads out of number of flips?

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Flipping a coin is a trial
 A “H” or “T” are outcomes of the trial.
 Sample Space = {H, T}
 P(H) = ½
 P(T) = ½

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Example
You have three nickels (coins) that you flip
into the air and onto the table.
 A trial would be flipping the three coins.
 One outcome would be heads-tails-heads
(HTH for short). What are the other
possible outcomes? (see next slide)
 An event is a subset of outcomes from a
sample space

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Example
 The sample space
for this trial of
flipping 3 coins (8 in this case.)
HHH
HTH
TTT
THT
HHT
HTT
TTH
THH
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Some Notation
Suppose we are interested in knowing all
the possible outcomes where we get two
heads and one tail. We will call this event
A:
 We can write A in set notation:

A =

{HHT, HTH, THH}
There are therefore 3 ways this can
happen out of a total of 8 possible
outcomes. We say the probability of the
event A happening is 3/8.
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For example, when rolling one die,
§ = {1,2,3,4,5,6}.
 Let the event E = rolling an even number.
Then the set E is all the ways to roll an
even number: E = {2,4,6}. This is a
subset of §.

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Theoretical Probability

Let E be an event. Then:
 P(E)
= k/n
k = the number of ways event E
can occur
 n = the total number of possible
outcomes


k
P( E ) 
n
*CAUTION: This formula is
only valid if each outcome is
equally likely.
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Example
A marble is drawn from a bag. There are
15 red, 12 yellow, and 18 blue marbles in
the bag.
 What is the probability of randomly
drawing a single red marble from the bag?
 What is the probability of randomly
drawing a single blue marble from the
bag?

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Question for Discussion
(something to think about. But we
will
discuss
later)from a bag. There are

A marble
is drawn
15 red, 12 yellow, and 18 blue marbles in
the bag.
 What is the probability of drawing a red
marble from the bag, setting it aside, and
then immediately drawing a second red
marble?
 Does the result change if you replace the
first red marble before drawing the second
time? If so, why?
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Empirical Probability
Empirical data is that which you observe.
For example, you have collected data that
indicates that of the last 550 loans a bank
granted, 42 of them were foreclosed upon.
 Then based on the empirical data, you
might say that the probability of a loan
going into foreclosure is 42/550.

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Empirical Probability
 Let
E be an event.
Then:
 P(E) = k/n
k = the number of
times event E has
occurred in the
past (under similar
circumstances)
 n = the number of
trials in the past

k
P( E ) 
n
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Law of Large Numbers

Empirical probabilities are basically just
estimates. They do not necessarily predict
the outcome of a particular trial. (“The next
loan will go to foreclosure!”) We do know
this, however: The outcome of one trial
cannot be predicted, but, one can predict
what will happen over a series of many trials.
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3-2: Combining Events
A marble is drawn from a bag. There are
15 red, 12 yellow, and 18 blue marbles in
the bag.
 What is the probability of randomly
drawing either a yellow or blue marble
from the bag?
 What about: “What is the probability of not
picking a red marble?”
 How are these related to the topics we
discussed in Chapter 2?

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3-3: Dice Problem

Suppose you have two dice that you roll
onto a table. Here is the sample space.
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Questions







What is the probability of getting a total of 10?
What is the probability of getting the same number on each
die?
What is the probability of getting a prime number total?
What is the probability of NOT getting a total of 7?
What is the probability of getting either a total of 8 or a total of
11?
What is the probability of getting either a 4 or an odd number
on one of the dies?
What is the probability of getting a 4 and an odd number on
the other die?
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Rolling a die
 Let event A = {an even number}
 Event B = {an odd number}
 Note that A and B are disjoint
 We call these events mutually exclusive
events when they don’t have any
outcomes in common.

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3-5: Rules of Probability





Let A and B be events, and let § be the
sample space.
Rule 1: 0  P(A)  1
Rule 2: P(§) = 1
Rule 3: P(AC) = 1 – P(A)
Rule 4: If A and B are mutually exclusive
events, then
 P(A
 B) = P(A) + P(B)
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Mutually exclusive versus
Independent events:
When you roll a die, the events { odd} and {even}
are mutually exclusive because they cannot
happen at the same time.
Hence, P(odd or even) = P(odd) +P(even)
When you roll a die two times, the events that the
1st die is odd and the 2nd die is odd are
independent events. The events does not affect
or influence each other.
Hence, Probability = P(odd) *P(odd) = 3/6*3/6
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Not mutually exclusive events
Roll a die….let us define
C = number that comes up is 1 or 2 or 3
D = number that comes up is 3 or 4 or 5
Hence, C and D are not mutually exclusive
P(C or D) = P(C) + P(D) - P(C and D)
= 3/6+3/6 - 1/6 = 5/6
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Rolling two dice:
 What
is the probability of getting a “4”
and an odd number on the other die?
Independent events:
hence, 1/6 * 3/6 = 3/36
{(4,1) (4,3), (4,5) }
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A marble is drawn from a bag. There are
15 red, 12 yellow, and 18 blue marbles in
the bag.
 What is the probability of drawing a red
marble from the bag, setting it aside, and
then immediately drawing a second red
marble? (not independent events)
 Does the result change if you replace the
first red marble before drawing the second
time? If so, why? (independent events)

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(a) 15/45 * 14/44
(b) 15/45 * 15/45
NOTE: Probability that the 2nd marble is red
knowing that the 1st marble is red and it is
not replaced is just 14/44
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3-6: Venn Diagrams
 A total
of 70 students are
randomly interviewed. 23 own a
car. 45 own a bike. 18 own both
a car and a bike. Draw a Venn
diagram that displays all of the
probabilities related to this survey.
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
(a)
(b)
(c)
(d)
Find the probabilities:
A student randomly chosen owns a car but not
a bike.
A student randomly chosen does not own
either a bike or a car.
A student randomly chosen owns a car or a
bike.
A student randomly chosen owns a car and a
bike.
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3-6B General Probability Formula

If A and B are events, then
 P(A

 B) = P (A) + P (B) – P (A  B)
Subtracting compensates for the doublecounting error.
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
Suppose 8% of a certain batch of
calculators have a defective case, and that
11% have defective batteries. Also, 3%
have both a defective case and defective
batteries. A calculator is selected from the
batch at random. Find the probability that
the calculator has a good case and good
batteries.
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
(a)
(b)
(c)
Ms Bezzone invites 10 relatives to a
party: her mother, 2 uncles, 3 brothers,
and 4 cousins. If the chances of any one
guest arriving first are equally likely, find
the probabilities:
The first guest is an uncle or a cousin.
The first guest is a brother or a cousin.
The first guest is an uncle or her mother.
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The table shows the probability of a person
accumulating credit card charges over a 12month period:
Charges
Probability
Under $100
0.31
$100-$499
0.18
$500-$999
0.18
$1000-$1999
0.13
$2000-$2999
0.08
$3000-$4999
0.05
$5000-$9999
0.06
$10000 or more
0.01
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Find the probability that a person’s total
charges during the period are
(a)
(b)
(c)
(d)
$500 or more
Less than $1000
$500 to $2999
$3000 or more
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Questions





For the following problems, the trial is rolling two
dice ( a red and a green die) (be sure to avoid
double-counting)
What is the probability of sum of both dice being
7?
What is the probability that the red die will show
an odd number or the sum of the two dice will be
8?
What is the probability that the green die is 6 or
the sum of the two dice is 10?
What is the probability that the green die shows
an even number and the sum of the two dice is
10?
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Group Exercise
See #36 from Chapter 3
 From a survey involving 1,000 people in a
certain city, it was found that 500 people
had tried a certain brand of diet cola, 600
had tried a certain brand of regular cola,
and 200 had tried both types of cola.
(Barnett p. 414)

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Group Exercise




Draw and label a Venn Diagram that demonstrates this
information in sets. Then try another Venn diagram
using the probabilities.
What is your sample space?
Find the probability that a randomly selected person
from the city has tried both of the colas. Find the
probability that a randomly selected person from the
city has tried the diet cola but not the regular cola.
Find the probability that a randomly selected person
from the city has tried the regular cola but not the diet
cola.
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Group Exercise



Find the probability that a randomly selected
person from the city has tried neither of the
colas.
Find the probability that a randomly selected
person from the city has tried either the diet or
the regular cola. Try to see if you can compute
this in TWO different ways.
Find the probability that a randomly selected
person from the city has tried one of the colas
but not both. Write your answer in probability
notation. Be careful on this one…a picture
should help.
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Focus on the Project

Let S be the event that an attempted work
out is successful and let F be the event
that it fails. Use the COUNTIF function to
find the fraction of past work outs which
were successful. This fraction is our
estimate for P(S). Likewise, we find the
fraction of attempts that failed and use this
as our estimate for P(F).
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Focus on the Project
Example 1:
 Use the Loan Records.xls to estimate
P(S).

 DCOUNT
will work
 COUNTIF will also do the job
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Focus on the Project



Go to the section titled Project 1 Specifics
(Chapter 7) and do Part 2a only.
Also do Chapter 3 Focus On Project Memo.
Edit your Written Report to reflect your newest
information. Be sure to use proper probability
and mathematics notation in your writing (use
the Equation Editor to format all mathematical
text).
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