Chapter 10 Probability

1
Chapter 10Probability
2
Experiments, Outcomes, andSample Space

• Outcomes Possible results from experiments in a
  random phenomenon
• Sample Space Collection of all possible outcomes
• S female, male
• S head, tail
• S 1, 2, 3, 4, 5, 6
• Event Any collection of outcomes
• Simple event event involving only one outcome
• Compound event event involving two or more
  outcomes

3
Basic Properties of Probability

• Probability of an event always lies between 0 1
• Sum of the probabilities of all outcomes in a
  sample space is always 1
• Probability of a compound event is the sum of the
  probabilities of the outcomes that constitute the
  compound event

4
Probability

• Equally Likely Events
• Probability as Relative Frequency
• Relative frequency ltgt Probability (Law of large
  numbers)
• Subjective Probability

5
Combinatorial Probability

• Using combinatorics to calculate possible number
  of outcomes
• Fundamental Counting Principle (FCP) Multiply
  each category of choices by the number of choices
• Combinations Selecting more than one item
  without replacement where order is not important
• Examples
• Lottery
• Dealing cards 3 of a kind

6
Marginal Probability

• The probability of one variable taking a specific
  value irrespective of the values of the others
  (in a multivariate distribution)
• Contingency table a tabular representation of
  categorical data

Purchased Warranty Did Not Purchase Warranty Total
Bought a used car 26 17 43
Bought a new car 73 35 108
Total 99 52 151
7
Conditional Probability

• The probability of an event occurring given that
  another event has already occurred

Purchased Warranty Did Not Purchase Warranty Total
Bought a used car 26 17 43
Bought a new car 73 35 108
Total 99 52 151
8
Conditional Probability
Purchased Warranty Did Not Purchase Warranty Total
Bought a used car 26 17 43
Bought a new car 73 35 108
Total 99 52 151
Event A Event B P(A) P(BA)
Used car Warranty 43/151.2848 26/43.6047
Used car No Warranty 43/151.2848 17/43.3953
New car Warranty 108/151.7152 73/108.6759
New car No Warranty 108/151.7152 35/108.3241
9
Conditional Probability
Purchased Warranty Did Not Purchase Warranty Total
Bought a used car 26 17 43
Bought a new car 73 35 108
Total 99 52 151
Event B Event A P(B) P(AB)
Warranty Used Card 99/151.6556 26/99.2626
Warranty New Car 99/151.6556 73/99.7374
No Warranty Used Card 52/151.3444 17/52.3269
No Warranty New Car 52/151.3444 35/52.6731
10
Joint of Events

• Set theory is used to represent relationships
  among events. In general, if A and B are two
  events in the sample space S, then
• A union B (A?B) either A or B occurs or both
  occur
• A intersection B (A?B) both A and B occur
• A is a subset of B (A?B) if A occurs, so does B
• A' or A event A does not occur (complementary)

11
Probability of Union of Events

• Mutually Exclusive Events if the occurrence of
  any event precludes the occurrence of any other
  events
• Addition Rule

12
Probability of Union of Events

• Probability of (bought a used car) or (purchased
  warrant)

Purchased Warranty Did Not Purchase Warranty Total
Bought a used car 26 17 43
Bought a new car 73 35 108
Total 99 52 151

• Probability of (Cr. Rating ? 700) or (Equity ?
  50)

Equity ? 50 Equity lt 50 Total
Cr. Rating ? 700 87 133 220
Cr. Rating lt 700 53 727 108
Total 140 860 1000
13
Probability of Mutually Exclusive Events

• Probability of (purchased warrant) or (Did not
  purchased warrant)

Purchased Warranty Did Not Purchase Warranty Total
Bought a used car 26 17 43
Bought a new car 73 35 108
Total 99 52 151

• Probability of (Cr. Rating ? 700) or (Cr. Rating
  lt 700)

Equity ? 50 Equity lt 50 Total
Cr. Rating ? 700 87 133 220
Cr. Rating lt 700 53 727 108
Total 140 860 1000
14
Probability of Complementary Events

• Complementary Events When two mutually exclusive
  events contain all the outcomes in the sample
  space

15
Probability of Intersection of Events

• Independent Events Event whose occurrence or
  non-occurrence is not in any way influenced by
  the occurrence or non-occurrence of another event
• Multiplication Rule

16
Probability of Intersection of Events
Purchased Warranty Did Not Purchase Warranty Total
Bought a used car 26 17 43
Bought a new car 73 35 108
Total 99 52 151
Event A Event B P(A) P(BA) P(A?B)
Used car Warranty 43/151.2848 26/43.6047 .1722
Used car No Warranty 43/151.2848 17/43.3953 .1126
New car Warranty 108/151.7152 73/108.6759 .4834
New car No Warranty 108/151.7152 35/108.3241 .2318
17
Probability of Intersection of Events
.1722
.2848
.1126
.4834
.2318
18
Probability of Intersection of Events
Purchased Warranty Did Not Purchase Warranty Total
Bought a used car 26 17 43
Bought a new car 73 35 108
Total 99 52 151
Event B Event A P(B) P(AB) P(A?B)
Warranty Used Card 99/151.6556 26/99.2626 .1722
Warranty New Car 99/151.6556 73/99.7374 .4834
No Warranty Used Card 52/151.3444 17/52.3269 .1126
No Warranty New Car 52/151.3444 35/52.6731 .2318
19
Probability of Intersection of Events
.1722
Warranty
.6556
.4834
.1126
No Warranty
.2318