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# Detailed Notes from Pre-Day Test STAT 110 001

USC

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This 8 page Class Notes was uploaded by Haley Johnson on Monday March 14, 2016. The Class Notes belongs to STAT 110 001 at University of South Carolina taught by Wendy Cimino in Spring 2016. Since its upload, it has received 72 views. For similar materials see Introduction to Statistical Reasoning in Statistics at University of South Carolina.

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Date Created: 03/14/16

Chapters 17 – 20 Notes Thinking About Chance • Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. • This is the basis for the idea of probability. Randomness and Probability • A phenomenon is random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions. • The probability of any outcome of a random phenomenon is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions. • An impossible outcome has probability of 0. • A certain outcome has probability of 1. Probability Terminology • The sample space, S, of an experiment is the collection of all possible outcomes. • An event is any collection of outcomes. An event may consist of one outcome or more than one outcome. 1 Chapters 17 – 20 Notes Proportions, Percentages, and Probabilities • A percentage is a proportion multiplied by 100%. • A probability is a proportion expressed as the long- run relative frequency of an outcome. • Proportions and probabilities are always between 0 and 1. • Percentages are always between 0% and 100%. Types of Probability • Experimental Probability – proportion of occurrences of a particular event when an experiment is repeated several times. • Theoretical Probability – For equally likely outcomes, = # ℎ # Example of a Random Phenomenon that Leads to Probabilities • Experiment – We flip two coins simultaneously and record the number of heads that occur. Possible Outcomes:0H 1H 2H Repeat experiment 200 times. Summary of Results: 0H 1H 2H 53 104 43 Calculate the experimental probability for the occurrence of 1H. What is the theoretical probability? 2 Chapters 17 – 20 Notes Myths About Chance • Short-run regularity • Example: What looks random? – Toss a coin 6 times. Which outcome is more probable? HTHTTH TTTHHH • Surprise Coincidence (Unusual Event) • college. Is this usual? London, you run into someone from • Law of Averages • Example: If you toss a coin six times and get TTTTTT, is the next toss more likely to be heads? Personal Probability • What is the probability that the Golden State Warriors will win the NBA championship this year? • A personal probability of an outcome is a number between 0 and 1 that expresses an individual’s judgment of how likely the outcome is. Example The probability that a randomly chosen driver will be involved in an accident in the next year is about 0.2. What do you think is your own probability of being in an accident in the next year? 3 Chapters 17 – 20 Notes Probability Models A probability model for a random phenomenon describes all possible outcomes and says how to assign probabilities to any collection of outcomes. Properties (Rules) of a Probability Model A:The probability of any event must be a number between 0 and 1. B:If we assign a probability to every possible outcome, the sum of these probabilities must equal 1. C:Complement Rule: The probability that an event does not occur is 1 minus the probability that the event does occur. A is the complement of A. c Rule C: P(A ) = 1 – P(A) Properties of a Probability Model, cont Mutually Exclusive Events have no outcomes in common. D:Addition Rule for Mutually Exclusive Events: If two events are mutually exclusive, the probability that one or the other occurs is the sum of their individual probabilities. If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B). 4 Chapters 17 – 20 Notes Example: Rolling a Die There are 6 outcomes and each outcome is equally likely to appear. The probability of any one outcome is 1/6. What is the probability of tossing an even number? What is the probability of tossing at most a 4? • Let A = the die lands on an even number. • Let B = the die lands on at most 4. • Are A and B mutually exclusive? Example Choose a student in a U.S. public high school at random and ask if he or she is studying a language other than English. Here is the distribution of the results (assume that no one takes more than one language): Language Spanish French German All others None Probability 0.08 0.02 0.03 0.57 Independence • Two events are independent if knowing the outcome of one does not change the probability for outcomes of the other. • When two events are independent, we find the probability of both events happening by multiplying their individual probabilities. If E and F are independent events, then P(E and F) = P(E)×P(F) 5 Chapters 17 – 20 Notes Example At a liberal arts college, 60% (probability = 0.6) of all freshmen are enrolled in a mathematics course, 73% are enrolled in an English course, and 49% are taking both. A freshman is randomly selected from this college. If M is the event that a freshman is taking a mathematics course and E is the event that a freshman is taking an English course, are M and E independent events? Simulation • Probability is based on many replications of an experiment. • Some experiments are difficult or impossible to replicate. • Simulation is often used to imitate chance behavior using a random digits table or computer software. Conducting a Simulation Step 1: Give a probability model Step 2: Assign digits to represent outcomes. Step 3: Simulate many replications (to estimate the probability of an event) 6 Chapters 17 – 20 Notes Simulation Example Toss a coin 10 times. What is the probability of a run of at least 3 consecutive heads or 3 consecutive tails? Step 1: P(Heads) = 0.5, P(Tails) = 0.5 Step 2: Odd digits represent heads – 1, 3, 5, 7, 9 Even digits represent tails – 0, 2, 4, 6, 8 Step 3: Simulate many repetitions Example A basketball player makes 70% of her free throws in a long season. If she takes 5 free throw shots in a game, what is the probability that she will miss 3 or more? The House Edge: Expected Values • The expected value of a random phenomenon that has numerical outcomes is found by multiplying each outcome by its probability and then adding all the products. • In symbols, if the possible outcomes are a , a , ..., a and their 1 2 k probabilities are1p 2 p , k.., p , then the expected value is Expected Value = a1p1+ a2p2+...+ k k • You can interpret expected value as the average value in the long run; over many trials. 7 Chapters 17 – 20 Notes Example Suppose X represents the number of glasses of soda that a randomly selected American drinks each day. The probability distribution for X is given below. X 0 1 2 3 4 5 Probability 0.55 0.28 0.09 0.04 0.03 0.01 Find the expected number of glasses of soda that anAmerican drinks each day. Note: Expected value does not have to be a possible outcome. Example The probability that a man aged 21 will die in the next year is about 0.0015. An insurance representative sells insurance to a 21-year-old male and charges $2000 for the policy which will pay $1,000,000if the individual dies. What is the expected profit for the representative on this po?icy Law of Large Numbers • If a random phenomenon with numerical outcomes is repeated many times independently, the mean of the actually observed outcomes approaches the expected value. • For theoretical probabilities, the expected value can be calculated exactly. • For experimentalprobabilities, we can find approximate expected values by simulation or experimentation. 8

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