chapter 10 probability. experiments, outcomes, and sample space outcomes: possible results from...
TRANSCRIPT
![Page 1: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/1.jpg)
Chapter 10
Probability
![Page 2: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/2.jpg)
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
![Page 3: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/3.jpg)
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
![Page 4: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/4.jpg)
Probability
• Equally Likely Events
• Probability as Relative Frequency
– Relative frequency <> Probability (Law of large numbers)
• Subjective Probability
nEP i
1)(
n
f
n
occursAtimesofNoAP
.)(
![Page 5: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/5.jpg)
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
![Page 6: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/6.jpg)
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
![Page 7: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/7.jpg)
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
![Page 8: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/8.jpg)
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
![Page 9: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/9.jpg)
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
![Page 10: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/10.jpg)
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)
![Page 11: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/11.jpg)
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
![Page 12: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/12.jpg)
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%)
![Page 13: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/13.jpg)
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)
![Page 14: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/14.jpg)
Probability of Complementary Events
• Complementary Events: When two mutually exclusive events contain all the outcomes in the sample space
0.1)()()()( APAPAorAPAAP
![Page 15: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/15.jpg)
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
![Page 16: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/16.jpg)
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
![Page 17: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/17.jpg)
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
![Page 18: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/18.jpg)
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
![Page 19: Chapter 10 Probability. Experiments, Outcomes, and Sample Space Outcomes: Possible results from experiments in a random phenomenon Sample Space: Collection](https://reader036.vdocuments.site/reader036/viewer/2022062423/56649f045503460f94c17bf1/html5/thumbnails/19.jpg)
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