Math310 Lecture Outline.docx

1、Math310 Lecture OutlineMath310 Lecture Outline.Chapter Two. The Basics of Probability.Vocabulary (Section 2.1). In yesterdays lab, we studied a situation that could be modeled by 11 black cards and 12 red cards that are shuffled and then 12 cards are dealt to Group A. Describe each of the following

2、for this situation.Experiment.Sample Space.Events. Simple Events. Compound Events.More Vocabulary. (Here A and B denote events).Union (A B)Intersection (A B)Complement (A or AC)Mutually Exclusive EventsDefinition of Probability (Section 2.2)Given an experiment with outcome space S, a probability is

3、a function P which assigns a real number P(A) to each event A S, subject to the following three rules: 1. 2. 3.Using probabilities to model. Probability representsEmpirical ProbabilityThe probability model.Interpretations of Probability.Objective ProbabilitySubjective ProbabilityNote. The mathematic

4、al development is limited to objective probability and caution should be taken when applying results to subjective probabilities. Other properties of the probability function. Any function P that satisfies the above three properties also satisfies some other useful properties.Property 1. All probabi

5、lities are between 0 and 1.Property 2. The complement rule.Property 3. The addition rule.Example 1. A person is applying for two jobs. They feel that the probability they will get an offer for the first job is 0.6, the probability they will get an offer for the second job is 0.8, and the probability

6、 they will get both offers is 0.5. a)Is this subjective or objective?b)Calculate the probability they will get at least one job offer.c)Calculate the probability they will get no job offers.Example 2. A certain system can have three different types of defects. Let Ai denote the event that the system

7、 has defect i. Suppose that P(A1)=0.12, P(A2)=0.07, P(A3)=0.05, P(A1A2)=0.06, P(A1A3)=0.03, P(A2A3)=0.02, and P(A1A2 A3)=0.01.a)Is this subjective or objective?b)Explain what the events A1A2 A3 and A1A2 A3 represent.c) Draw a Venn diagram representing the probabilities. d)What is the probability tha

8、t a system does not have a defect?e)What is the probability that a system has at least one defect?f)What is the probability that a system has at least two defects?g)What is the probability that a system has all three defects?Counting and Probability (Section 2.3). Suppose a sample space S contains f

9、initely many elements, and assume that each of these outcomes is equally likely. In this case, given any event A S, we can calculate the probability that A occurs in this way:Multiplication Principle and Tree Diagrams.Example 3a)How many 1 topping pizzas can be made if you have 3 choices for size, 2

10、 choices for crust type, and 12 pizza toppings to choose from?b)How many pizzas can be made if you can include any number of toppings?Permutations.Example 4. If there are 12 players on a little league baseball team, how many ways can the 9 field positions be assigned?Combinations.Example 5. A commit

11、tee of 10 would like to create a subcommittee of 3. How many different ways can this be done?Hyper-geometric Probabilities. Setting Logic FormulaExample 6. Fifteen telephones have just been received at an authorized service center (5 cellular, 5 cordless, and 5 corded). Suppose that the phones are s

12、erviced in random order.a)What is the probability that all of the cellular phones are among the first 6 to be serviced?b)What is the probability that after servicing 6 phones, phones of only two of the three types remain to be serviced?c)What is the probability that after servicing 10 phones, phones

13、 of only two of the three types remain to be serviced?d)What is the probability that two phones of each type are among the first 6 to be serviced?Example 7 (Section 2.4). A shopkeeper phones to tell you that she has two new baby beagles. You ask her if she has a female puppy, and she replies that sh

14、e does. However, on the way to meet her, you change your mind and decide that you would rather have a male puppy. What is the probability that she has a male puppy?a)Guess the answer.b)Do a simulation using pennies. Record the empirical probability from the simulation using results of whole class.c)

15、Give original sample space and probability assignment and reduced sample space and probability assignment. Use this to give the exact probability.d)Notation.Conditional Probability. Intuition via Venn Diagrams. Formal Definition.Example 8. Suppose you roll two fair six-sided dice, one of which is re

16、d and one of which is blue. You record the numbers showing on each die. Let A = “the blue die comes up 3”, B = “at least one of the dice comes up 3”, and C = “the sum of the dice is 7”. Calculate the following:a)P( A | B )b)P( B | A )c)P( A | C )d)P( C | A)e)P( B | C )f)P( C | B ).Multiplication Rul

17、e.Example 9. Suppose our experiment is to draw 2 cards from a standard deck without replacement and record the sequence of cards selected (an ordered sample without replacement). Let B = “the first card is a Spade” and let A = “the second card is a Spade”a)Find P(A|B) and P(A|B)b)Compute P(AB) and P

18、(AB).Example 10. Suppose you know that exactly 4 light bulbs out of a box of 10 bulbs are defective. You plan to locate the defective bulbs by drawing the bulbs out one at a time and testing them (without replacement). Find the probability thata)You find the 4th defective bulb on or before the seven

19、th test.b)You find the 4th defective bulb on the seventh test.Working with Conditionals.For some experiments you can calculate P(AB) and P(B) . This can be used to find P( A|B ) . What examples did we do this on?Other times you can calculate P(B) and P( A|B ). This can be used to find P(AB) . What e

20、xamples did we do this on?Example 11. Suppose that a person is selected at random from a large population of which 1% are drug users and that a drug test is administered that is 98% reliable. If a person tests positive for drugs, what is the probability that they are a drug user?a)What do you think

21、the answer is?b)Introduce Notation.c)Simulation via Maple.d)Calculation.Independent Events (Section 2.5). Events A and B are said to be independent if P ( A | B ) = P(A). In other words, “if we know that B occurs, it does not change the probability that A occurs.”Multiplication Rule for Independent

22、Events. P( A B ) = P(A) P(B).Notes.1)If events A and B are independent, it follows that P ( B | A ) = P(B). This can be shown using definition of independence, conditional probability, and (general) multiplication rule. (Thus, the roles of A and B are symmetric in the definition of independence.)2)I

23、f A and B are independent events, then the following pairs of events are also independent:a)A and B b)A and Bc)A and BDetermining Dependence or IndependenceExample 12. Consider experiment “Draw one card from a standard deck.”a)Let A = “the card is an ace” and B = “the card is a heart”.Are A and B in

24、dependent?b)Let A = “the card is the ace of hearts” and B = “the card is a heart”. Are A and B independent?Example 13. Consider the experiment “Roll two six sided die.” a)Let A = “Roll a sum of 5” and B = “The first die is a three”. Are A and B independent?b)Let A = “Roll a sum of 7” and B = “The fi

25、rst die is a three”. Are A and B independent?Example 14. a)Draw two cards from a standard deck without replacement, and record the sequence of cards drawn. Let A = “the second card is a spade”and B = “the first card is a spade”.Are A and B independent?b)What is sampling was done with replacement?Usi

26、ng independence to calculate probability.Example 15. Consider an electrical circuit with components A, B, C, and D. Each operates independently and is 90% reliable. Components A and B are connected in series as are components C and D; these two units are then connected in parallel. Find the probabil

27、ity that the electrical circuit works.Example 16. The probability that a grader will make a marking error on any particular question of a true false exam is 0.1. There are 10 questions and questions are marked independently.a)What is the probability that no errors are made?b)What is the probability

28、that exactly two errors are made?Generalization of Example 16. Suppose instead that there are n true false questions and the probability of making a marking error is p.a)What is the probability that no errors are made?b)What is the probability that exactly two errors are made?c)What is the probabili

29、ty that exactly k errors are made?Binomial Probabilities. Setting Logic (previous example) FormulaThe Last Example. A 10 question true false test needs to be re-graded if any of the questions are graded incorrectly. (Questions are graded independently of each other, each with the same probability of

30、 graded incorrectly.) a)If 15% of all tests need to be re-graded, what is the probability that a question is graded incorrectly?b)How small should the probability of mis-grading a question be to ensure that only 5% of exams need to be re-graded?Summary of Chapter 2Counting Rules Multiplication Princ

31、iple Permutations CombinationsProbability Rules Always between 0 and 1 Rule for Complements. Addition Rule For Mutually Exclusive Events. General Addition Rule Multiplication Rule for Independent Events General Multiplication RuleConditional Probabilities Find Conditional Probabilities from those of single events and intersections. Find Probabilities of intersections from those of single events and conditionals.Independent Events Use mathematical definition to check if events are independent. Assume events are independent and calculate probabi