Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Announcements Finite Probability Friday, September 30th I MyMathLab 4 is due Wednesday Oct 5 I Problem Set 4 is due Friday Oct 7 Today: Sec. 6.1: Experiments, Outcomes, Sample Spaces, Events Describe the sample space and events of an experiment Use set operations to define new events Next Class: Sec. 6.2: Assignment of Probabilities Cherveny Sept 30 Math 1004: Probability Definitions Definition I An experiment is an activity with an observable result. I An outcome is a possible result of an experiment. I The sample space of the experiment is the set of all possible outcomes. I An event is a subset of the sample space. Cherveny Sept 30 Math 1004: Probability Committee Example Example A committee of two people is to be selected from five people, Renee, Stacy, Travis, Ursula, and Victor (R,S,T,U,V). (a) What is the sample space for this experiment? Answer: {RS, RT , RU, RV , ST , SU, SV , TU, TV , UV } (b) Describe the event “Renee is on the committee” as a subset of the sample space. Answer: {RS, RT , RU, RV } (c) Describe the event “neither Renee nor Stacy is on the committee” as a subset of the sample space. Answer: {TU, TV , UV } Cherveny Sept 30 Math 1004: Probability Events and Sets Since events are subsets of the sample space (our “universe” of outcomes), we may use set operations to define new events. I Union: E1 ∪ E2 occurs when E1 or E2 (or both) occur I Intersection: E1 ∩ E2 occurs when both E1 and E2 occur I Complement: E 0 occurs when E does not Cherveny Sept 30 Math 1004: Probability Coin Toss Example An experiment consists of tossing a coin three times and observing the sequence of heads and tails. (a) What is the sample space of this experiment? (b) Determine the event E1 = “more heads than tails occur” (c) Determine the event E2 = “the first toss is a head” (d) Determine the event E1 ∩ E2 (e) Determine the event E10 (f) Determine the event (E1 ∪ E2 )0 Cherveny Sept 30 Math 1004: Probability Practice 1. A coin is tossed and a die is rolled, and both results are observed. How many outcomes are in the sample space? 2. A letter is selected at random from the word “MISSISSIPPI” (a) What is the sample space for this experiment? (b) Describe the event “the letter chosen is a vowel” as a subset of the sample space. 3. An orange and a black six-sided die are rolled, and the results facing up are observed. (a) List the sample space, or just write out a few examples and say how many elements there are. (b) Describe each of the following events: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) Cherveny Both numbers are even. At least one number is even. Neither number is less than or equal to 3. The sum of the numbers is 6. The sum of the numbers is greater than or equal to 8. The numbers are the same. A 2 or 3 occurs, but not both 2 and 3. No 4 appears. Sept 30 Math 1004: Probability Practice Answers 1. 2 · 6 = 12 outcomes 2. (a) {M, I , S, P} (b) {I } 3. (a) {(1, 1), (1, 2) . . . , (1, 6), (2, 1), (2, 2), . . . , . . . , (6, 6)}. There are 6 · 6 = 36 outcomes in the sample space. (b) Cherveny (i) There are 9 outcomes in this event. When a problem asks you to describe or list outcomes, you should give a set not just a size... i.e. {(2, 2), (2, 4), (2, 6), (4, 2), . . . } (ii) There are 27 outcomes in this event. (iii) There are 9 outcomes in this event. (iv) There are 5 outcomes in this event. (v) There are 15 outcomes in this event. (vi) There are 6 outcomes in this event. (vii) There are 18 outcomes in this event. (viii) There are 25 outcomes in this event. Sept 30 Math 1004: Probability