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Copyright © 2012 Pearson Canada Inc., Toronto, Ontario
PROBABILITY
Slide 1- 1
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 2
Dealing with Random Phenomena
 A random phenomenon is a situation in which we know
what outcomes could happen, but we don’t know which
particular outcome did or will happen.
 In general, each occasion upon which we observe a
random phenomenon is called a trial.
 At each trial, we note the value of the random
phenomenon, and call it an outcome.
 When we combine outcomes, the resulting combination is
an event.
 The collection of all possible outcomes is called the
sample space.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 3
First a definition . . .
 When thinking about what happens with
combinations of outcomes, things are simplified if
the individual trials are independent.
 Roughly speaking, this means that the
outcome of one trial doesn’t influence or
change the outcome of another.
 For example, coin flips are independent.
The Law of Large Numbers
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 4
The Law of Large Numbers (cont.)
 The Law of Large Numbers (LLN) says that the
long-run relative frequency of repeated
independent events gets closer and closer to a
single value.
 We call the single value the probability of the
event.
 Because this definition is based on repeatedly
observing the event’s outcome, this definition of
probability is often called empirical probability.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 5
Theoretical Probability
 When probability was first studied, a group of French
mathematicians looked at games of chance in which all
the possible outcomes were equally likely.
 It’s equally likely to get any one of six outcomes from
the roll of a fair die.
 It’s equally likely to get heads or tails from the toss of a
fair coin.
 However, keep in mind that events are not always equally
likely.
 A skilled basketball player has a better than 50-50
chance of making a free throw.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 6
 The probability of an event is the number of
outcomes in the event divided by the total
number of possible outcomes.
P(A) =
Modeling Probability (cont.)
# of outcomes in A
# of possible outcomes
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 7
The First Three Rules of Working with Probability
 We are dealing with probabilities now, not data,
but the three rules don’t change.
 Make a picture.
 Make a picture.
 Make a picture.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 8
The First Three Rules of Working with
Probability (cont.)
 The most common kind of picture to make is
called a Venn diagram.
 We will see Venn diagrams in practice shortly…
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 9
Formal Probability
1. Two requirements for a probability:
 A probability is a number between 0 and 1.
 For any event A, 0 ≤ P(A) ≤ 1.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 10
Formal Probability (cont.)
2. Probability Assignment Rule:
 The probability of the set of all possible
outcomes of a trial must be 1.
 P(S) = 1 (S represents the set of all possible
outcomes.)
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 11
Formal Probability (cont.)
3. Complement Rule:
 The set of outcomes that are not in the event
A is called the complement of A, denoted AC
.
 The probability of an event occurring is 1
minus the probability that it doesn’t occur:
P(A) = 1 – P(AC
)
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 12
Formal Probability (cont.)
4. Addition Rule:
 Events that have no outcomes in common
(and, thus, cannot occur together) are called
disjoint (or mutually exclusive).
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 13
Formal Probability (cont.)
4. Addition Rule (cont.):
 For two disjoint events A and B, the
probability that one or the other occurs is the
sum of the probabilities of the two events.
 P(A or B) = P(A) + P(B), provided that A and
B are disjoint.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 14
Formal Probability
5. Multiplication Rule (cont.):
 For two independent events A and B, the
probability that both A and B occur is the
product of the probabilities of the two events.
 P(A and B) = P(A) x P(B), provided that A
and B are independent.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 15
Formal Probability (cont.)
5. Multiplication Rule (cont.):
 Two independent events A and B are not
disjoint, provided the two events have
probabilities greater than zero:
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 16
Formal Probability (cont.)
5. Multiplication Rule:
 Many Statistics methods require an
Independence Assumption, but assuming
independence doesn’t make it true.
 Always Think about whether that assumption
is reasonable before using the Multiplication
Rule.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 17
Formal Probability - Notation
Notation alert:
 In this text we use the notation P(A or B) and
P(A and B).
 In other situations, you might see the following:
 P(A ∪ B) instead of P(A or B)
 P(A ∩ B) instead of P(A and B)
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 18
Putting the Rules to Work
 In most situations where we want to find a
probability, we’ll often use the rules in
combination.
 A good thing to remember is that sometimes it
can be easier to work with the complement of the
event we’re really interested in.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 19
The General Addition Rule
 When two events A and B are disjoint, we can
use the addition rule for disjoint events from
Chapter 14:
P(A or B) = P(A) + P(B)
 However, when our events are not disjoint, this
earlier addition rule will double count the
probability of both A and B occurring. Thus, we
need the General Addition Rule.
Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 20
The General Addition Rule (cont.)
 General Addition Rule:
 For any two events A and B,
P(A or B) = P(A) + P(B) – P(A and B)
 The following Venn diagram shows a situation in
which we would use the general addition rule:

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Basics of probability

  • 1. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario PROBABILITY Slide 1- 1
  • 2. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 2 Dealing with Random Phenomena  A random phenomenon is a situation in which we know what outcomes could happen, but we don’t know which particular outcome did or will happen.  In general, each occasion upon which we observe a random phenomenon is called a trial.  At each trial, we note the value of the random phenomenon, and call it an outcome.  When we combine outcomes, the resulting combination is an event.  The collection of all possible outcomes is called the sample space.
  • 3. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 3 First a definition . . .  When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent.  Roughly speaking, this means that the outcome of one trial doesn’t influence or change the outcome of another.  For example, coin flips are independent. The Law of Large Numbers
  • 4. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 4 The Law of Large Numbers (cont.)  The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to a single value.  We call the single value the probability of the event.  Because this definition is based on repeatedly observing the event’s outcome, this definition of probability is often called empirical probability.
  • 5. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 5 Theoretical Probability  When probability was first studied, a group of French mathematicians looked at games of chance in which all the possible outcomes were equally likely.  It’s equally likely to get any one of six outcomes from the roll of a fair die.  It’s equally likely to get heads or tails from the toss of a fair coin.  However, keep in mind that events are not always equally likely.  A skilled basketball player has a better than 50-50 chance of making a free throw.
  • 6. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 6  The probability of an event is the number of outcomes in the event divided by the total number of possible outcomes. P(A) = Modeling Probability (cont.) # of outcomes in A # of possible outcomes
  • 7. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 7 The First Three Rules of Working with Probability  We are dealing with probabilities now, not data, but the three rules don’t change.  Make a picture.  Make a picture.  Make a picture.
  • 8. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 8 The First Three Rules of Working with Probability (cont.)  The most common kind of picture to make is called a Venn diagram.  We will see Venn diagrams in practice shortly…
  • 9. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 9 Formal Probability 1. Two requirements for a probability:  A probability is a number between 0 and 1.  For any event A, 0 ≤ P(A) ≤ 1.
  • 10. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 10 Formal Probability (cont.) 2. Probability Assignment Rule:  The probability of the set of all possible outcomes of a trial must be 1.  P(S) = 1 (S represents the set of all possible outcomes.)
  • 11. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 11 Formal Probability (cont.) 3. Complement Rule:  The set of outcomes that are not in the event A is called the complement of A, denoted AC .  The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(AC )
  • 12. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 12 Formal Probability (cont.) 4. Addition Rule:  Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint (or mutually exclusive).
  • 13. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 13 Formal Probability (cont.) 4. Addition Rule (cont.):  For two disjoint events A and B, the probability that one or the other occurs is the sum of the probabilities of the two events.  P(A or B) = P(A) + P(B), provided that A and B are disjoint.
  • 14. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 14 Formal Probability 5. Multiplication Rule (cont.):  For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events.  P(A and B) = P(A) x P(B), provided that A and B are independent.
  • 15. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 15 Formal Probability (cont.) 5. Multiplication Rule (cont.):  Two independent events A and B are not disjoint, provided the two events have probabilities greater than zero:
  • 16. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 16 Formal Probability (cont.) 5. Multiplication Rule:  Many Statistics methods require an Independence Assumption, but assuming independence doesn’t make it true.  Always Think about whether that assumption is reasonable before using the Multiplication Rule.
  • 17. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 17 Formal Probability - Notation Notation alert:  In this text we use the notation P(A or B) and P(A and B).  In other situations, you might see the following:  P(A ∪ B) instead of P(A or B)  P(A ∩ B) instead of P(A and B)
  • 18. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 18 Putting the Rules to Work  In most situations where we want to find a probability, we’ll often use the rules in combination.  A good thing to remember is that sometimes it can be easier to work with the complement of the event we’re really interested in.
  • 19. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 19 The General Addition Rule  When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14: P(A or B) = P(A) + P(B)  However, when our events are not disjoint, this earlier addition rule will double count the probability of both A and B occurring. Thus, we need the General Addition Rule.
  • 20. Copyright © 2012 Pearson Canada Inc., Toronto, Ontario Slide 14- 20 The General Addition Rule (cont.)  General Addition Rule:  For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B)  The following Venn diagram shows a situation in which we would use the general addition rule: