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Chapter 10 Probability
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Chapter 10
Probability
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
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
1)(0 EP
1)( EP
Probability
• Equally Likely Events
• Probability as Relative Frequency
– Relative frequency <> Probability (Law of large numbers)
• Subjective Probability
nEP i
1)(
n
f
n
occursAtimesofNoAP
.)(
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
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
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
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(B|A)
Used carWarranty
43/151=.284826/43=.6047
No Warranty 17/43=.3953
New carWarranty
108/151=.715273/108=.6759
No Warranty 35/108=.3241
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(A|B)
WarrantyUsed Card
99/151=.655626/99=.2626
New Car 73/99=.7374
No WarrantyUsed Card
52/151=.344417/52=.3269
New Car 35/52=.6731
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 (AB) = either A or B occurs or both occur
– A intersection B (AB) = both A and B occur
– A is a subset of B (AB) = if A occurs, so does B
– A' or Ā = event A does not occur (complementary)
Probability of Union of Events
• Mutually Exclusive Events: if the occurrence of any event precludes the occurrence of any other events
• Addition Rule
n
ii
n
ii EPEP
11
)()(
)()()()( 212121 EEPEPEPEEP
)()()()(
)()()()(
ABCPBCPACPABP
BPBPAPCBAP
Probability of Union 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
• Probability of (bought a used car) or (purchased warrant)
Equity 50% Equity < 50% Total
Cr. Rating 700 87 133 220
Cr. Rating < 700 53 727 108
Total 140 860 1000
• Probability of (Cr. Rating 700) or (Equity 50%)
Probability of Mutually Exclusive 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
• Probability of (purchased warrant) or (Did not purchased warrant)
Equity 50% Equity < 50% Total
Cr. Rating 700 87 133 220
Cr. Rating < 700 53 727 108
Total 140 860 1000
• Probability of (Cr. Rating 700) or (Cr. Rating < 700)
Probability of Complementary Events
• Complementary Events: When two mutually exclusive events contain all the outcomes in the sample space
0.1)()()()( APAPAorAPAAP
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
n
ii
n
ii EPEP
11
)()(
)|()()|()()( BAPBPABPAPBAP
)()|( APBAP
)()()( BPAPBAP
)()|( BPABP
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(B|A) P(AB)
Used carWarranty
43/151=.284826/43=.6047 .1722
No Warranty 17/43=.3953 .1126
New carWarranty
108/151=.715273/108=.6759 .4834
No Warranty 35/108=.3241 .2318
Warranty
No Warranty
.6759
.3241
Warranty
No Warranty
.6047
.3953
Used Car
New Car
.7152
Probability of Intersection of Events
.2848
.1722
.1126
.4834
.2318
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(A|B) P(AB)
WarrantyUsed Card
99/151=.655626/99=.2626 .1722
New Car 73/99=.7374 .4834
No Warranty
Used Card52/151=.3444
17/52=.3269 .1126
New Car 35/52=.6731 .2318
Used Car
New Car
.2626
.7374
Used Car
New Car
.3269
.6731
.3444
Probability of Intersection of Events
.6556
.1722
.1126
.4834
.2318
Warranty
No Warranty
