Lesson 5 Probability 1 What are Probabilities 2 2 Finding Probabilities 3 3 Conditional Probabilities 10 4 Applying Probability Rules 13 www apsu edu jonesmatt 1 1 What are Probabilities There are different kinds Subjective probabilities The chance I pass this course is probably 80 The chance I finish my homework on time is 50 Experimental probabilities Sally won Rock Paper Scissors 25 times out of 71 plays so the experimental probability she wins a game is 25 71 These are usually based on counting the number of observations of an event divided by the total number of trials Theoretical probabilities Theoretically the chance Sally wins a game of Rock Paper Scissors is 1 3 These probabilities are determined by reasoning Chapters 5 and 6 focus on theoretical probabilities www apsu edu jonesmatt 2 2 Finding Probabilities An experiment is an action whose outcome cannot be predicted with certainty An experiment will result in some outcome The sample space of the experiment is the collection of all possible outcomes of an experiment An event is a result of an experiment and is a subset of the sample space It is a collection of outcomes from the sample space A Venn diagram is sort of a cartoon of events in a sample space www apsu edu jonesmatt 3 Given two events A and B their intersection A B consists of all points in both A and B Their union A B consists of all points in either A or B The complement AC of the event A consists of all points not in A www apsu edu jonesmatt 4 Example 1 Let A be the event a randomly chosen student is an agriculture major and B be the event the student is a business major Interpret and construct Venn diagrams for the following events A B A B www apsu edu jonesmatt AC A B C AC B C A B C 5 Suppose an experiment has N equally likely outcomes Let E be an event meaning a subset of the set of outcomes and let f f for frequency be the number of outcomes resulting in E The probability of the event E written P E is the number of ways E can occur divided by the total number of outcomes N f P E N The probability of any event is always a real number from the unit interval 0 1 The closer a probability of an event is to 1 the more likely the event is to occur The closer a probability of an event is to 0 the less likely the event is to occur The probability of an event that must occur is 1 The probability of an event that cannot occur is 0 www apsu edu jonesmatt 6 Example 2 Suppose you roll a red and a green die Find 1 P sum is 11 2 P doubles 3 P sum is 4 4 P sum is at most 4 5 P at least one die shows 6 6 P the red die shows a 6 7 P both dice show 6 8 P sum of dice exceeds 1 9 P sum of dice exceeds 12 10 P exactly one die shows 4 3 www apsu edu jonesmatt 7 Probability Axioms 1 The probability of any event is a number in the unit interval 0 1 2 The probability of getting some event in the sample space is 1 3 For disjoint or mutually exclusive events E1 E2 En meaning Ei Ej for all i 6 j P E1 E2 En P E1 P E2 P En In general for two events A and B we have the general addition rule P A B P A P B P A B If A and B are disjoint then P A B P A P B since because then P A B P 0 The rule for complements is P AC 1 P A www apsu edu jonesmatt 8 Example 3 For the Venn diagram shown suppose P A 0 2 P B 0 12 and P C 0 38 Calculate P A B C Example 4 For the Venn diagram shown suppose P A 0 2 P B 0 12 P C 0 38 P B C 0 10 Calculate P A B C and P A B C www apsu edu jonesmatt 9 3 Conditional Probabilities P A B P A B P B Example 5 Suppose 2 of students are double majoring in agriculture and business and 3 of students are majoring in business Given that a randomly chosen student majors in business what is the probability s he also majors in agriculture www apsu edu jonesmatt 10 Independent Events Two events A and B are independent if the probability that A occurs given that B has occurred is the same as the probability of A P A B P A 1 Statement 1 is equivalent to the statements P B A P B and P A B P A P B NOTE INDEPENDENCE AND DISJOINTNESS ARE NOT THE SAME www apsu edu jonesmatt 11 Example 6 Suppose you toss a fair die four times What is the probability you see 5 5 6 then 1 What is the probability that you see four even numbers in a row What is the probability you get a straight Example 7 You draw one card from a standard deck of 52 Are the events that the card is an ace and the card is a spade independent Example 8 Each time you play your friend in backgammon you win with probability 2 3 independently of the other outcomes What is the probability that you win at least twice in three games Example 9 It is independently rainy on Monday with probability 0 2 rainy on Wednesday with probability 0 3 and rainy on Thursday with probability 0 4 What is the chance that it rains Monday and Thursday but not Wednesday www apsu edu jonesmatt 12 4 Applying Probability Rules Example 10 How many people should be in a room for there to be at least a 50 chance that two or more share the same birthday What is the chance that two or more people share the same birthday in your class Example 11 Random Drug Testing of Air Traffic Controllers p 250 www apsu edu jonesmatt 13