Probability Theory: Understanding Random Events and Calculating Probabilities, Study notes of Data Analysis & Statistical Methods

Download Probability Theory: Understanding Random Events and Calculating Probabilities and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! Chapter 5 Basic Probability  Gaming industry sets odds for lotteries based on probability theory  Card Games (gambling)  Sporting Events (horse racing)  Weather Forecast Real-Life Probabilities  Random Phenomenon: An event for which the outcome is a priori (beforehand) unknown  Example: ◦ # emails we receive today ◦ An election outcome ◦ Tomorrow’s temperature  Notation: ◦ Random event in question: Capital Letter (X or Y) ◦ Observed event: Lowercase letter (x or y) Section 5.1: Definitions  Probability: methodology that quantifies the likelihood of outcomes for random phenomena ◦ Long-run proportion of times the outcome of interest would occur for a particular event  Many times we repeat the same random event over and over again to look at the long-run proportion  Trials: set of repetitions of the same random event n times, where n is the sample size Denoted: nxxx ,,, 21   Independent Trials: the outcome of one trial has no influence on another trial ◦ Example: Rolling 2 dice  Law of Large Numbers (LLN): As n increases for independent trials we expect:  Sample mean to approach the Population mean  Sample proportion to approach the population proportion x ppˆ 6. Does the coin seem fair? What if there were 12 heads? 13? 14? 6? 7. If the coin is fair and we continue flipping indefinitely, LLN suggests that the proportion of heads would approach what value? HH TH TT TH HT TH HH TT HT HH  Writing out the sequence of events for trials is lengthy  We still have no way to make decisions on things like “Is the coin unfair?”  Goals For Future: ◦ Summarize our phenomena with numeric variables so we can use our chapter 2 skills ◦ Use probabilities to quantify the idea of “unlikely” What we have seen so far…  Sample Space (S): lists all possible outcomes, or elements  Event: Any subset of S  Complement: all the elements of S that are not part of the event Section 5.2: Calculating Probabilities  Mutually Exclusive: Two events that have no elements in common A:event B:event A B S Empty set and BA 1. What are the possible outcomes? 2. -event that contains odd numbers -event that contains even numbers 3. Are and disjoint? i.e. Ex. 5.2: Consider rolling a Die. 1E 2E 2E1E 21 EE   Probability: methodology that quantifies the likelihood of outcomes for random phenomena  Probability of event A denoted by P(A) Notation of Probability Sin outcomes of # AEvent in outcomes of # )( AP 5. If A and B are independent (outcome of one trial as no effect on the outcome of another trial), The Multiplicative Rule The Axioms of Probability )()()( BPAPBAP   Consider a family with 2 kids. M=male F=female 1. List the possible gender outcomes for 2 kids. 2. If B=at least one male child is born, what is P(B)? Ex: 5.3 3. If G=at least one female child is born, what is the P(G)? 4. If N=exactly 1 female born, what is the P(N) 5. What is ? ◦ )( BGNP   Conditional Probability: probability of events given that certain events have occurred ◦ Similar to conditional proportions calculated from contingency tables in Chapter 3  The Conditional Probability of B given A  The Conditional Probability of A given B Section 5.3: Conditional Probabilities Give n )( )( )|( AP BAP ABP   )( )( )|( BP BAP BAP    A math teacher gave her class two tests. 16% of the class passed both tests and 46% of the class passed the first test. What percent of those who passed the first test also passed the second test? A-Passed the first test B-Passed the first and second test Ex. 5.5 Ex. 5.6  Suppose a study of speeding violations and drivers who use car phones produced the following fictional data: Speeding Violation in the last year No speed violation in the last year Totals Car Phone User 25 280 305 Not a Car Phone User 45 405 450 Total 70 685 755 4. P(person is a car phone user OR person had no violation in the last year) 5. P(person is a car phone user GIVEN person had a violation in the last year) 6. P(person had no violation last year GIVEN person was not a car phone user)  Recall: Two events are independent if  This implies that conditional probabilities for independent events are Independence )()()( BPAPBAP  )()|( )()|( APBAP BPABP   Proof of Independence: )( )( )()( )( )( )|( BP AP BPAP AP ABP ABP  