Chapter 7 Probability

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Chapter 7 Probability

• 7.1 Experiments, Sample Spaces, and Events
• 7.2 Definition of Probability
• 7.3 Rules of Probability
• 7.4 Use of Counting Techniques in Probability
• 7.5 Conditional Probability and Independent
  Events
• 7.6 Bayes Theorem

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Section 7.1 Experiments, Sample Spaces, and
Events
An experiment is an activity with observable
results (called outcomes).
A sample point is an outcome of an experiment.
The sample space is the set of all possible
sample points.
An event is a subset of a sample space.
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Ex. Rolling a die
Outcomes landing with a 1, 2, 3, 4, 5, or 6 face
up. Sample Space S 1, 2, 3, 4, 5, 6 Events
Certain event
Impossible event
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Ex. An experiment consists of spinning the hand
on the disk below two times. Find the sample
space.
P
C
W
S (P,C), (P,W), (P,P), (C,P), (C,W), (C,C),
(W,P), (W,C), (W,W)
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Events
The union of events A and B is the event
The intersection of events A and B is the event
The complement of event A is the event AC.
Ex. Rolling a die. S 1, 2, 3, 4, 5, 6 Let
A 1, 2, 3 and B 1, 3, 5
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Events A and B are mutually exclusive if
Ex. When rolling a die, if event A 2, 4, 6
(evens) and event B 1, 3, 5 (odds), then A
and B are mutually exclusive.
Ex. When drawing a single card from a standard
deck of cards, if event A heart, diamond
(red) and event B spade, club (black), then
A and B are mutually exclusive.
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Section 7.2 Definition of Probability
The probability of an event occurring is a
measure of the proportion of the time that the
event will occur in the long run.
Suppose that in n trials an event E occurs m
times. The relative frequency of the event E is
m/n. If the relative frequency approaches some
value P(E) as n becomes larger, then P(E) is
called the empirical probability of E.
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Ex. The table below represents the frequency of
certain types of license plates observed by a
family on a recent trip. Find the probability
distribution.
State Number
Wisconsin 45
Illinois 80
Iowa 20
Indiana 5
Probability
45/150 0.300
80/150 0.533
20/150 0.133
5/150 0.033
Notice 150 total observations
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Let S s1, s2, s3,,sn where each si
represents a simple event (all mutually
exclusive) and let P(si) represent the
probability of event si.
The function P, which assigns a probability to
each simple event is called a probability
function.
Also P(si) has the following properties
probabilities are between 0 and 1
Sum of the probabilities is 1
Probabilities of the union is the sum of their
probabilities
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Probability of an Event in a Uniform Sample Space

• If
• S s1, s2,
  , sn
• is the sample space for an experiment in which
  the outcomes are equally likely, then we assign
  the probabilities
• to each of the outcomes s1, s2, , sn.

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Ex. Assume that when rolling a die each face is
equally likely to show up. If event E 2 then
since S 1, 2, 3, 4, 5, 6, we have P(E) 1/6.
That is, the probability of rolling a 2 is 1 in
6.
Similarly, the probability of rolling any face
number is 1/6.
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Finding the Probability of an Event E

• Determine a sample space S associated with the
  experiment.
• Assign probabilities to the simple events of S.
• If E s1, s2, s3,,sn (each a simple event)
  then
• P(E) P(s1) P(s2) P(sn).
  
• If E is the empty set then P(E) 0.

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Ex. An experiment consists of spinning the hand
on the disk below one time. Assume each outcome
is equally likely.
A
C
W
Find P(C) and then find
Notice S C, A, W each of which has a
probability of 1/3.
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Applied Example Rolling Dice

• A pair of fair dice is rolled.
• Calculate the probability that the two dice show
  the same number.
• Calculate the probability that the sum of the
  numbers of the two dice is 6.

Applied Example 3, page 365
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Applied Example Rolling Dice

• Solution
• The sample space S of the experiment has 36
  outcomes
• S (1, 1), (1, 2), , (6, 5), (6, 6)
• Both dice are fair, making each of the 36
  outcomes equally likely, so we assign the
  probability of 1/36 to each simple event.
• The event that the two dice show the same number
  is
• E (1, 1), (2, 2) , (3, 3), (4, 4), (5, 5), (6,
  6)
• Therefore, the probability that the two dice show
  the same number is given by

Six terms
Applied Example 3, page 365
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Applied Example Rolling Dice

• Solution
• The event that the sum of the numbers of the two
  dice is 6 is given by
• E6 (1, 5), (2, 4) , (3, 3), (4, 2), (5, 1)
• Therefore, the probability that the sum of the
  numbers on the two dice is 6 is given by

Applied Example 3, page 365
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Section 7.3 Rules of Probability
Properties of the Probability Function
If E and F are mutually exclusive (E ? F Ø),
then
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Ex. A local grocery store has found kept track
of the amount of money spent by its customers on
a single visit. Find the probability that if a
customer is selected at random, the amount spent
by the customer will be

1. More than 150
2. More than 50 but less than or equal to 200

0.15
0.50
Dollars spent Probability
0.05
0.10
0.15
0.25
0.45
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Property 4 Addition Rule
If E and F are any two events of an experiment,
then
Subtract overlap
E
F
Note If E and F are mutually exclusive, then
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Ex. A card is drawn from a well-shuffled deck of
52 playing cards. What is the probability that
it is a king or a heart?
K King and H Heart
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Property 5 Rule of Complements
If E is an event of an experiment and EC denotes
the complement of E, then
Ex. A card is drawn from a well-shuffled deck of
52 playing cards. What is the probability that
it is not a king?
K pick a king,
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Section 7.4 Use of Counting Techniques in
Probability
Computing the Probability of an Event in a
Uniform Sample Space
Let S be a uniform sample space and let E be any
event. Then
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Ex. Suppose that you reach into a box of 12 size
AA batteries and you know that 4 of them are
dead. Find the probability that
a. in one draw you get a good battery.
b. in two draws without replacement you get two
good batteries.
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Ex. Three balls are selected at random without
replacement from the jar below. Find the
probability that
a. All 3 of the balls are green.
b. One ball is red and two are black.
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Ex. Refer to the jar of marbles below. Two
marbles are drawn at random without replacement.
Find the probability that no yellow are drawn.
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Section 7.5 Conditional Probability and
Independent Events
The probability of an event is affected by the
knowledge of other information relevant to the
event.
Notation P(AB) is read the probability of
event A given that event B has occurred.
Ex. You roll a fair die. Find the probability
that you roll a 2 given that your roll is an even.
Knowing it is even restricts the sample space to
2, 4, 6.
So
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Conditional Probability of an Event
If A and B are events in an experiment and
then the conditional probability that
the event B will occur given that A has already
occurred is
Which can be written (the Product Rule)
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Ex. In a box of 20 size AA batteries, 10 are
brand X and 10 are brand Y. You also know that 3
of the brand X batteries are dead, while 2 of the
brand Y are dead. Find the probability that in a
(random) draw
a. you get a dead (D) brand X battery.
b. you get brand Y given that you drew a dead (D)
battery.
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Independent Events
If A and B are independent events, then
Test for Independent Events
Events A and B are independent events if and only
if
Note this generalizes to more than two
independent events.
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Ex. If A die is rolled twice, show that rolling
a 5 on the first roll and rolling a 4 on the
second roll are independent events.
(roll 1, roll2)
V 5 on first roll, R 4 on second roll
Therefore V and R are independent
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Section 7.6 Bayes Theorem
Bayes Theorem
Let A1, A2, , An be a partition of a sample
space S and let E be an event of the experiment
such that P(E) is not zero. Then the posteriori
probability P(AiE) is given by
Where
Posteriori probability probability is
calculated after the outcomes of the experiment
have occurred.
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Ex. A store stocks light bulbs from three
suppliers. Suppliers A, B, and C supply 10,
20, and 70 of the bulbs respectively. It has
been determined that company As bulbs are 1
defective while company Bs are 3 defective and
company Cs are 4 defective. If a bulb is
selected at random and found to be defective,
what is the probability that it came from
supplier B?
Let D defective
So about 0.17