1. Chapter 10 Probability

2. Experiments, Outcomes, and Sample 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 <> 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 car261743 Bought a new car7335108 Total9952151

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 car261743 Bought a new car7335108 Total9952151

8. Conditional Probability Purchased Warranty Did Not Purchase Warranty Total Bought a used car261743 Bought a new car7335108 Total9952151 Event AEvent BP(A)P(B|A) Used car Warranty 43/151=.2848 26/43=.6047 No Warranty17/43=.3953 New car Warranty 108/151=.7152 73/108=.6759 No Warranty35/108=.3241

9. Conditional Probability Purchased Warranty Did Not Purchase Warranty Total Bought a used car261743 Bought a new car7335108 Total9952151 Event BEvent AP(B)P(A|B) Warranty Used Card 99/151=.6556 26/99=.2626 New Car73/99=.7374 No Warranty Used Card 52/151=.3444 17/52=.3269 New Car35/52=.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 Ā = 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 Purchased Warranty Did Not Purchase Warranty Total Bought a used car261743 Bought a new car7335108 Total9952151 Probability of (bought a used car) or (purchased warrant) Equity  50% Equity < 50%Total Cr. Rating  700 87133220 Cr. Rating < 70053727108 Total1408601000 Probability of (Cr. Rating  700) or (Equity  50%)

13. Probability of Mutually Exclusive Events Purchased Warranty Did Not Purchase Warranty Total Bought a used car261743 Bought a new car7335108 Total9952151 Probability of (purchased warrant) or (Did not purchased warrant) Equity  50% Equity < 50%Total Cr. Rating  700 87133220 Cr. Rating < 70053727108 Total1408601000 Probability of (Cr. Rating  700) or (Cr. Rating < 700)

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 car261743 Bought a new car7335108 Total9952151 Event AEvent BP(A)P(B|A) P(A  B) Used car Warranty 43/151=.2848 26/43=.6047.1722 No Warranty17/43=.3953.1126 New car Warranty 108/151=.7152 73/108=.6759.4834 No Warranty35/108=.3241.2318

17. 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

18. Probability of Intersection of Events Purchased Warranty Did Not Purchase Warranty Total Bought a used car261743 Bought a new car7335108 Total9952151 Event BEvent AP(B)P(A|B) P(A  B) Warranty Used Card 99/151=.6556 26/99=.2626.1722 New Car73/99=.7374.4834 No Warranty Used Card 52/151=.3444 17/52=.3269.1126 New Car35/52=.6731.2318

19. 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