Chapter 10 Probability

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

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

Probability Equally Likely Events Probability as Relative Frequency –Relative frequency <> Probability (Law of large numbers) Subjective Probability

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 Bought a new car Total

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 Bought a new car Total

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

Conditional Probability Purchased Warranty Did Not Purchase Warranty Total Bought a used car Bought a new car Total Event BEvent AP(B)P(A|B) Warranty Used Card 99/151= /99=.2626 New Car73/99=.7374 No Warranty Used Card 52/151= /52=.3269 New Car35/52=.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 (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)

Probability of Union of Events Mutually Exclusive Events: if the occurrence of any event precludes the occurrence of any other events Addition Rule

Probability of Union of Events Purchased Warranty Did Not Purchase Warranty Total Bought a used car Bought a new car Total Probability of (bought a used car) or (purchased warrant) Equity  50% Equity < 50%Total Cr. Rating  Cr. Rating < Total Probability of (Cr. Rating  700) or (Equity  50%)

Probability of Mutually Exclusive Events Purchased Warranty Did Not Purchase Warranty Total Bought a used car Bought a new car Total Probability of (purchased warrant) or (Did not purchased warrant) Equity  50% Equity < 50%Total Cr. Rating  Cr. Rating < Total 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

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

Probability of Intersection of Events Purchased Warranty Did Not Purchase Warranty Total Bought a used car Bought a new car Total Event AEvent BP(A)P(B|A) P(A  B) Used car Warranty 43/151= /43= No Warranty17/43= New car Warranty 108/151= /108= No Warranty35/108=

Warranty No Warranty Warranty No Warranty Used Car New Car.7152 Probability of Intersection of Events

Probability of Intersection of Events Purchased Warranty Did Not Purchase Warranty Total Bought a used car Bought a new car Total Event BEvent AP(B)P(A|B) P(A  B) Warranty Used Card 99/151= /99= New Car73/99= No Warranty Used Card 52/151= /52= New Car35/52=

Used Car New Car Used Car New Car Probability of Intersection of Events Warranty No Warranty