Probability

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Chapter 5 Probability Section 5.1 Probability of
Simple Events
Probability is a measure of the likelihood of a
random phenomenon or chance behavior.
Probability describes how likely it is that some
event will occur.

• Probability falls into 3 major approaches.
• Classical Approach
• Empirical/Experimental Approach
• Subjective Approach
• We will discuss each approach in detail, but
  first we need to look at some basic ideas
  associated with probability.

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Probability vs. Statistics
Population known
Population unknown
Infer sample composition
Take sample and infer population
In probability, an experiment is any process that
can be repeated in which the results are
uncertain. Probability experiments do not always
produce the same results or outcome, so the
result of any single trial of the experiment is
not known ahead of time.
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Suppose we are to flip a coin one time, what is
the probability that we observe a tails?
½, 0.5, 50/50, 50
So if we flip the coin 10 times, would we
definitely see 5 tails? Why not?
What if we flipped the coin 100 times? A million
times?
The Law of Large Numbers As the number of
repetitions of a probability experiment
increases, the proportion with which a certain
outcome is observed gets closer to the
probability of the outcome.
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Suppose I have a fair die and I am going to roll
that die one time and observe the outcome. What
are all the possible outcomes?
The sample space, S, of a probability experiment
is the collection of all possible outcomes or
simple events.
S 1,2,3,4,5,6
A simple event is any single outcome from a
probability experiment. Each simple event is
denoted ei.
e1 1, e2 2, , e6 6,
An event is any collection of outcomes from a
probability experiment. An event may consist of
one or more simple events. Events are denoted
using capital letters such as E.
E roll an odd 1,3,5 F roll a 1,2,3
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Properties of Probabilities We define the
probability of an event, denoted P(E) , as the
likelihood of that event occurring.
Probabilities have some properties that must be
satisfied. 1. The probability of any event E,
P(E), must be between 0 and 1 inclusive. That
is, 0 ? P(E) ? 1. 2. If an event is impossible,
the probability of the event is 0. 3. If an
event is a certainty, the probability of the
event is 1. 4. If S e1, e2, . . . , en then
P(e1) P(e2) . . . P(en) 1. We will
now discuss the three methods or approaches for
determining probabilities.
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Classical Approach The classical method of
computing probabilities requires equally likely
outcomes. An experiment is said to have equally
likely outcomes when each simple event has the
same probability of occurring. Some examples
would be each number of a die, each card in a
deck of cards, and each side of a coin.
Computing Probabilities Using the Classical
Method If an experiment has n equally likely
simple events and if the number of ways that an
event E can occur is m, then the probability of
E, P(E), is
So, if S is the sample space of this experiment,
then
where N(E) is the number of simple events in E
and N(S) is the number of simple events in the
sample space.
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Example Let the sample space be S
1,2,3,4,5,6,7,8,9,10. Suppose the simple
events are equally likely. Compute the
probability of the event E an odd number
E 1,3,5,7,9
N(E) 5 N(S) 10
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Empirical/Experimental Approach In this
approach, probabilities are obtained from
empirical evidence, that is, evidence based upon
the outcomes of a probability experiment.
Approximating Probabilities through the Empirical
Approach The probability of an event E is
approximately the number of times event E is
observed divided by the number of repetitions of
the experiment. P(E) ? relative frequency of E
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Example On September 8, 1998, Mark McGwire hit
his 62nd homerun of the season. Of the 62
homeruns he hit, 26 went to left field, 21 went
to left center, 12 went to center, 3 went to
right center and 0 went to right field.

• What is the probability that a randomly selected
  homerun was hit to left center field?
• What is the probability that a randomly selected
  homerun was hit to right field?
• Is it impossible for Mark McGwire to hit a
  homerun to right field?

P(left ctr) 21/62 0.34
P(Rt) 0/62 0
No.
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Tree Diagrams Tree diagrams can be used to
determine the sample space of an experiment.
Example Compute the probability of having one
boy and three girls in a four-child family
assuming boys and girls are equally likely.
START
0.5
0.5
1st Child
B
G
0.5
0.5
0.5
0.5
2nd Child
G
G
B
B
3rd Child
G
G
G
B
B
B
B
G
4th Child
G
B
B
B
G
G
B
G
B
G
B
G
B
G
B
G
1 path P(B,G,G,G) (0.5)(0.5)(0.5)(0.5) 0.54
0.0625
4 pathways 0.0625 0.0625 0.0625 0.0625
4(0.0625) 0.25
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Subjective Approach Subjective probabilities are
probabilities obtained based upon an educated
guess. If you watch the Weather Channel, maybe
they say that the chance of rain today is 50,
but the local news says that there is a 75
chance of rain today. These are very different
chances for rain. The reason for these
differences is because people interpret
information differently. Because subjective
probabilities are based upon personal judgments,
they should be interpreted with extreme
skepticism.
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Section 5.2 The Addition Rule Complements
Compound Events are formed by combining two or
more simple events.
1) The probability that both events E and F will
occur P(E and F) P(E?F)
Intersection
2) The probability that either E or F will occur
P(E or F) P(E?F)
Union
3) The probability that event E will occur given
that event F has already occurred. P(EF)
read as probability of E given F
Conditional Probability
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• Let E and F be two events.
• E and F is the event consisting of simple events
  that belong to both E and F.
• E or F is the event consisting of simple events
  that belong to either E or F or both.

Example Let E 1,2 F 2,3
E n F 2
E U F 1,2,3
Let us visualize these concepts using Venn
Diagrams.
S
E
F
E n F
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Addition Rule For any two events E and F, P(E
or F) P(E?F) P(E) P(F) P(E and F) P(E)
P(F) P(E?F)
S
E
F
Counted twice
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If events E and F have no simple events in common
or cannot occur simultaneously, they are said to
be disjoint or mutually exclusive.
Addition Rule for Mutually Exclusive Events If E
and F are mutually exclusive events, then P(E or
F) P(E) P(F)
S
E
F
Note E n F Ø thus, P(E n F) 0
P(E or F) P(E) P(F) P(E?F)
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Complements Suppose the probability of an event E
is known and we would like to determine the
probability that E does not occur. This can
easily be accomplished using the idea of
complements.
Complement of an Event Let S denote the sample
space of a probability experiment and let E
denote an event. The complement of E, denoted
EC, is all simple events in the sample space S
that are not simple events in the event E.
Complement Rule If E represents any event and EC
represents the complement of E, then P(EC) 1
P(E)
S
Ec
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• Example
• A standard deck of cards contains 52 cards. One
  card is randomly selected from the deck.
• Compute the probability of randomly selecting a
  two or three from a deck of cards.
• Compute the probability of randomly selecting a
  two or three or four from a deck of cards.
• Compute the probability of randomly selecting a
  two or club from a deck of cards.
• Compute the probability of randomly selecting a
  card other than a two from a deck of cards.

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• Compute the probability of randomly selecting a
  two or three from a deck of cards.

2. Compute the probability of randomly selecting
a two or three or four from a deck of cards.
Note Mutually exclusive events as above.
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3. Compute the probability of randomly selecting
a two or club from a deck of cards. 4.
Compute the probability of randomly selecting a
card other than a two from a deck of cards.
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Section 5.3 The Multiplication Rule
Conditional Probability The notation P(F E) is
read the probability of event F given event E.
It is the probability of an event F occurring
given the occurrence of the event E.
The Multiplication Rule The probability that two
events, E and F both occur is P(E and F) P(E n
F) P(E) P(F E) In words, the probability
of E and F is the probability of event E
occurring times the probability of event F
occurring given the occurrence of event E.
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Example Let S 1,2,3,4 E 1,2 F
2,3 then (EnF) 2 What is P(EnF) ?
S
E
F
1
2
3
4
Note P(EnF) is referred to as the joint
probability of E and F. P(E) is referred to as
the marginal probability of E.
Also note Conditional
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Example A bag of 30 tulip bulbs was purchased
from a nursery. The bag contains 12 red tulip
bulbs, 10 yellow tulip bulbs and 8 purple tulip
bulbs.
First draw a tree diagram

• What is the probability that two randomly
  selected tulip bulbs will both be red?
• What is the probability that the first bulb
  selected is red and the second is yellow?
• What is the probability that the first bulb
  selected is yellow and the second is red?
• What is the probability that one bulb is red and
  the other yellow?

12/30
8/30
10/30
11/29
8/29
12/29
7/29
10/29
12/29
8/29
10/29
9/29

• What is the probability that two randomly
  selected tulip bulbs will both be red?
• What is the probability that the first bulb
  selected is red and the second is yellow?
• What is the probability that the first bulb
  selected is yellow and the second is red?
• What is the probability that one bulb is red and
  the other yellow? Tree on board

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• What is the probability that two randomly
  selected tulip bulbs will both be red?
• What is the probability that the first bulb
  selected is red and the second is yellow?
• What is the probability that the first bulb
  selected is yellow and the second is red?
• What is the probability that one bulb is red and
  the other yellow?

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Two events E and F are independent if the
occurrence of event E in a probability experiment
does not affect the probability of event F.
Two events are dependent if the occurrence of
event E in a probability experiment affects the
probability of event F.
From previous example P(R)12/30 ? P(RY)
12/29
Independent Events Two events E and F are
independent if and only if P(F E) P(F) or
P(E F) P(E)
Multiplication Rule for Independent Events If E
and F are independent events, the probability
that E and F both occur is P(E and F) P(E n
F) P(E)P(F) In words, the probability of E
and F is the probability of event E occurring
times the probability of event F occurring.
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Example 2 fair coins P(E head) 0.5 P(F
tail) 0.5 Each toss is an independent
event. So P(E n F) P(E)P(F) 0.25
Mutually Exclusive vs. Independent
P(EF) P(E)
P(E U F) P(E) P(F)
S
S
E
F
E
F
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Section 5.4 Conditional Probability
Conditional Probability Rule If E and F are any
two events, then
The probability of event F occurring given the
occurrence of event E is found by dividing the
probability of E and F by the probability of E.
Or, the probability of event F occurring given
the occurrence of event E is found by dividing
the number of simple events in E and F by the
number of simple events in E.
Likewise,
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Hence, we can use Bayes Rule to conclude,
P(EF)P(F) P(FE)P(E)
P(E n F) i.e. Multiplication Rule
Example A box contains 100 microchips, some of
which were produced by factory 1 and the rest by
factory two. Some of the chips are defective and
some are good. An experiment consists of
choosing one microchip at random from the box and
testing whether it is good or defective. The
data are presented in the following table.
Like a contingency table, but is this probability
or statistics?
20 80 100
60 40
Population is known, therefore probability
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Make a joint probability table
Joint Probabilities
Marginal Probabilities

• Find the probability of being defective.
• Find the probability of being made in factory
  one.
• Find the probability of being good.
• Find the probability of being made in factory
  two.
• Find the probability of being defective and made
  in factory one.
• Find the probability of being defective given
  made in factory one.
• Find the probability of made in factory one
  given defective.
• Are the events of selecting a defective chip and
  one made at factory one independent events?
• Are the events of selecting a defective chip and
  one made at factory one mutually exclusive
  events?

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• Find the probability of being defective.
• Find the probability of being made in factory
  one.
• Find the probability of being good.
• Find the probability of being made in factory
  two.

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5. Find the probability of being defective and
made in factory one. 6. Find the
probability of being defective given made in
factory one. 7. Find the probability of made
in factory one given defective.
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8. Are the events of selecting a defective chip
and one made at factory one independent
events? 9. Are the events of selecting a
defective chip and one made at factory one
mutually exclusive events?
No. P(DFac1) 0.25 ? 0.20 P(D)
No. P(DnFac1) 0.15 ? 0
Note Joint Tables may be constructed by two
means. - Empirical - Theoretical