Transcript PPT Chapter 15 - McGraw Hill Higher Education

Introductory Mathematics
& Statistics
Chapter 15
Elementary Probability
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Learning Objectives
• Understand elementary probability concepts
• Calculate the probability of events
• Distinguish between mutually exclusive, dependent and
independent events
• Calculate conditional probabilities
• Understand and use the general addition law for
probabilities
• Understand and apply Venn diagrams
• Understand and apply probability tree diagrams
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15.1 Introduction
• In everyday language we often refer to the probability that certain
events will happen
• We also use the word ‘chance’ as a substitute for probability on
some occasions
• While we all use the word ‘probability’ in our language, there
would be few people who could provide a formal definition of its
meaning
Examples
– There is a 10% chance that it will rain
– There is a 30% chance that Essendon will win the AFL
premiership in the year 2010
– There is a 25% chance that a certain investment will yield a profit
in the coming year
– There is a 50–50 chance that I will get a tax refund next year
– The probability that a 767 jet plane will crash into the Sydney
Harbour Bridge before the year 2030 is 1 in 100 million
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15.2 Probability of events
•
Sample space
– When a statistical experiment is conducted, there are a
number of possible outcomes
– These possible outcomes are called a sample space and
this is denoted by S
 E.g. a coin is tossed. What is the sample space?
 Solution: S = {head, tail}
•
Events
– An event is a specified subset of a sample space.
 E.g. a coin is tossed. Define event A as the outcome
‘heads’
 Solution: A = outcome is a head
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15.2 Probability of events (cont…)
• Events (cont…)
– More than one event can be defined from a sample
space.
 E.g. suppose a card is drawn at random from a pack of
52 playing cards. Define events A, B and C as drawing
an ace, red card and face card, respectively
 Solution: A = card drawn is an ace, B = card drawn is
red, C = card drawn is a face card
– The impossible event (or empty set) is one that contains
no outcomes. It is often denoted by the Greek letter 
(phi)
 E.g. a hand of 5 cards is dealt from a deck z. Let A be
the event that the hand contains 5 aces. Is this
possible?
 Solution : Since there are only 4 aces in the deck, event
A cannot occur. Hence A is an impossible event.
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15.2 Probability of events (cont…)
• Probability
– If A is an event, the probability that it will occur is denoted by
P(A)
– The probability (or chance) that an event A will occur is the
proportion of possible outcomes in the sample space that yield
the event A. That is:
P  A 
number of outcomes that yield event A
total number of possible outcomes
– The definition makes sense only if the number of possible
outcomes (the sample space) is finite
– If an event can never occur, its probability is 0. An event that
always happens has probability 1
– The value of a probability must always lie between 0 and 1
– A probability may be expressed as a decimal or a fraction
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15.2 Probability of events (cont…)
• Mutually exclusive events
– Two events A and B are said to be mutually exclusive if they
cannot occur simultaneously
– If two events A and B are mutually exclusive, the following
relationship holds:
PA or B  PA   PB
– Suppose that A1, A 2 , A 3 , An are mutually exclusive
events. Then:
PA1 or A 2 or A3  An 
 PA1   PA 2   PA3     PAn 
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15.2 Probability of events (cont…)
• Independent events
– Two events A and B are independent events if the occurrence of
one does not alter the likelihood of the other event occurring
– Events that are not independent are called dependent events
– If two events A and B are independent, the following relationship
holds:
PAandB   PA   PB
– Suppose that A1, A 2 , A 3 , An are n independents events. Then
PA1 and A 2 and A3 and An 
 PA1  PA 2   PA3     PAn 
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15.2 Probability of events (cont…)
• Complementary events
– The complement of an event is the set of outcomes of a
sample space for which the event does not occur
– Two events that are complements of each other are said to
be complementary events
(Note: complementary events are mutually exclusive)
– Suppose we define the events:
A = no one has the characteristic
B = at least 1 person has the characteristic
Then A and B are complementary events
P (at least 1 person has the characteristic)
= 1 – P (no person has the characteristic)
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15.2 Probability of events (cont…)
• Conditional probabilities
– The probability that event A will occur, given that an event
B has occurred, is called the conditional probability that A
will occur, given that B has occurred
– The notation for this conditional probability is P (A|B)
– For any two events, A and B, the following relationship
holds:
PA and B
PA B 
PB
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15.2 Probability of events (cont…)
• Conditional probabilities (cont…)
– If two events A and B are independent
PA B  PA   PB
– Substituting this result
PA   PB
PA B 
PB
 PA 
– That is, for independent events A and B the conditional
probability that event A will occur, given that event B had
occurred, is simply the probability that event A will occur
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15.2 Probability of events (cont…)
• The general addition law
– When two events are not mutually exclusive, use the
following general addition law
PA or B  PA   PB  PA and B
– If the events A and B are mutually exclusive,
P(A and B) = 0
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15.3 Venn diagrams
• Sample spaces and events are often presented in a visual
display called a Venn diagram
• Use the following conventions
– A sample space is represented by a rectangle
– Events are represented by regions within the rectangle. This
is usually done using circles
• Venn diagrams are used to assist in presenting a picture of
the union and intersection of events, and in the calculation of
probabilities
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15-13
15.3 Venn diagrams (cont…)
• Definitions
– The union of two events A and B is the set of all outcomes that
are in event A or event B. The notation is:
Union of event A and event B = A ∪ B
Hence, we could write, for example, P (A ∪ B) instead of
P(A or B)
– The intersection of two events A and B is the set of all
outcomes that are in both event A and event B. The notation
is:
Intersection of event A and event B = A ∩ B
Hence, we could write, for example, P (A ∩ B) instead of
P(AandB)
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15.3 Venn diagrams (cont…)
The shaded area is event A
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15.3 Venn diagrams (cont…)
• The union of two events A and B is the set of all
outcomes that are in event A or event B
Union of event A and event B  A  B
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15.3 Venn diagrams (cont…)
• The intersection of two events A and B is the set of all
outcomes that are in both event A and event B.
Intersecti on of event A and event B  A  B
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15.3 Venn diagrams (cont…)
• The intersection of events A, B and C is the set of all
outcomes that is in events A, B and C
Intersecti on of events A and B and C  A  B  C
B
A
C
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15-18
15.4 Probability tree diagrams
• Probability tree diagrams can be a useful visual display
of probabilities
• The diagrams are especially useful for determining
probabilities involving events that are not independent
• The joint probabilities for combinations of these events
are found by multiplying the probabilities along the
branches from the beginning of the tree
• If the events are not independent, the probabilities on
the second tier of branches will be conditional
probabilities, since their values will depend on what
happened in the first event
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15.4 Probability tree diagrams (cont..)
• Example
– A clothing store has just imported a new range of suede
jackets that it has advertised at a bargain price on a rack
inside the store. The probability that a customer will try on a
jacket is 0.40. If a customer tries on a jacket, the probability
that he or she will buy it is 0.70. If a customer does not try
on a jacket, the probability that he or she will buy it is 0.15.
– Calculate the probability that:
(a) a customer will try on a jacket and will buy it
(b) a customer will try on a jacket and will not buy it
(c) a customer will not try on a jacket and will buy it
(d) a customer will not try on a jacket and will not buy it
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15.4 Probability tree diagrams (cont..)
Solution
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Summary
• We have looked at understanding elementary
probability concepts
• We calculated the probability of events
• We distinguished between mutually exclusive,
dependent and independent events
• We also looked at calculating conditional probabilities
• We understood and used the general addition law for
probabilities
• We understood and applied Venn diagrams
• We understood and applied probability tree diagrams
Copyright  2010 McGraw-Hill Australia Pty Ltd
PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
15-22