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Math 011 – CHAPTER 4 Probability DEFINITION Event Simple event Sample space Example: Procedure Example of Event NOTATION: Finding Probabilities: 1. Relative Frequency Approximation of Probability 2. Classical Approach to Probability (Requires equally likely outcomes) 3. Subjective Probability Sample Space Law of Large Numbers Example: (Relative Frequency Probability) A recent survey of 1010 adults in the U.S showed 202 of them smoke. Find the probability that a randomly selected adult in the U.S. smokes. Example: (Classical Approach) Assuming that boys are equally likely, find the probability of getting children of the same gender. Example: (Subjective Probability) a. Find the probability of our Statistics class being cancelled. b. Find the probability that you’ll get stuck in the next elevator that you ride. Rare Event Rule for Inferential Statistics: Caution: Example: Bills from 17 large cities in the U.S. were analyzed for presence of cocaine. Results: 23 of the bills were not tainted and 211 were tainted by cocaine. Based on results, find the probability that a randomly selected bill is tainted by cocaine. Example: If a year is randomly selected, find the probability that Thanksgiving Day in the U.S. is on a) Wednesday; b) Thursday Complement 202 Example: (Complement of a smoker) We found earlier that P(smoker) = 1010 = .20. Find the probability of randomly selecting an adult in the U.S. who does not smoke. Rounding off Probabilities Example: a. b. c. d. e. Probability of 0.8208356 Probability of 0.0357469 Probability of 2/3 Probability of 4/8 Probability of 0.0002546 Interpreting Probabilities An event is unlikely An event has an unusually low number An event has an unusually high number Example: a. A fair coin is tossed 1,000 times and exactly 500 heads result. Probability of getting 500 heads in 1000 tosses is 0.0252. Is this result unlikely? Is 500 heads unusually low or unusually high? b. A fair coin is tossed 1,000 times and 10 heads result. Is this result unlikely? Is 10 heads unusually low or unusually high? 4-3 ADDITION RULE Compound Event Notation: Example: Pre-Employment Drug Testing – Find the probability of selecting a subject who had a negative test result or doesn’t use drugs. Subject Uses Drugs Subject Doesn’t Use Drugs Formal addition Rule Intuitive Addition Rule Events A and B are disjoint Example: Disjoint event Not disjoint events: Positive Test Result 44 90 Negative Test result 6 860 Rule of Complementary Events Example: According to a survey, 19.8% of college students take at least 1 online class. What’s the probability of randomly selecting a college student who doesn’t take any online courses? Example: Pre-employment Drug Testing – Find the probability of selecting a subject who had a positive test result or uses drugs. Subject Uses Drugs Subject Doesn’t Use Drugs Positive Test Result 44 90 Negative Test result 6 860 4-4 MULTIPLICATION RULE: BASICS Notation: Formal Multiplication Rule Caution: Two events A and B are independent Two events are dependent Example: Subject Uses Drugs Positive Test Result 44 Negative Test result 6 a. If two of the 50 subjects who used drugs are randomly selected with replacement, find the probability that the first had a positive result and the second had a negative result. b. If two of the 50 subjects who used drugs are randomly selected without replacement, find the probability that the first had a positive result and the second had a negative result. Example: Among respondents asked which their favorite seat on a plane is, 492 said window seat, 8 chose the middle seat, and 306 chose the aisle seat. a. What is the probability of randomly selecting one of the people and getting one who did not choose the middle seat? b. If 2 are randomly selected without replacement, what is the probability that neither chose the middle seat? c. If 25 different people are randomly selected without replacement, what is the probability that none chose the middle seat? Example: (BIRTHDAYS) When 2 different people are randomly selected from your class, find the indicated probability by assuming birthdays occur on the days of the week with equal frequency. a. Probability that two people are born on the same day of the week. b. Probability that the two people are both born on a Monday. 4-5 MULTIPLICATION RULE Complements and Conditional Probability Find the probability of getting at least one of some event: Example: Find the probability of getting at least 7 when four digits are randomly selected with replacement for a lottery ticket. Conditional Probability Example: Subject Uses Drugs Subject Doesn’t Use Drugs Positive Test Result 44 90 Negative Test result 6 860 a. If one of the 1000 subjects is randomly selected, find the probability that the subject had a positive result, given the subject actually uses drugs. b. Find P(subject uses drugs | positive test results) Confusion of the Inverse: 4-6 PERMUTATIONS AND COMBINATIONS Does order count? DEFINTIONS Permuatations Combinations Notation: COUNTING RULES: 1. Fundamental Counting Rule Example: How many different codes are there in a 2-character code consisting of a letter and one digit? In a 3-code with a letter and two digits? 2. Factorial Rule: Example: a. Find the number of ways that 3 different letters can be arranged? b. Then find the number of arrangements for 5 different letters. 1. Permutations Rule (when all items are different) Example: If 4 letters {a,b,c,d} are available and two are selected without replacement, find the number of different permutations. 2. Permutations Rule ( when some items e the same as others) Example: If 8 letters are available {a,a,b,b,c,c,c,c} and all are selected without replacement, what is the number of different permutations? 3. Combination Rule Example: If four letters {a,b,c,d} are available and two are selected without replacement, how many different combinations are there? Example: How many different ways can you touch three or more fingers to each other on one hand?