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Chapter 10 Introducing Probability Random = individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions Basic Vocabulary of Probability Probability = the proportion of times the outcome would occur in a very long series of repetitions ◦ We can never observe a probability exactly ◦ Mathematical probability is an idealization based on imagining what would happen in an indefinitely long series of trials ◦ The proportion in a small or moderate number of tosses can be far from the probability. Probability describes long runs only. Basic Vocabulary of Probability Probability of getting heads should be: _______ Flip 10 times. How many heads did you get? _______ What is your proportion of heads? _______ Compare to the proportion of heads gotten for the entire class. Example: Flipping a coin 250 #10.4 Probability is a measure of how likely an event is to occur. Match One of the probabilities that follow with each statement. 0 0.01 0.3 0.6 0.99 1 a. This even is impossible. It can never occur. b. This event is certain. It will occur on every trial. c. This event is very unlikely, but it will occur once in a while in a long sequence of trials. d. This event will occur more often than not. Sample space = S = the set of possible outcomes Event = an outcome or a set of outcomes of a random phenomenon. ◦ An event is a subset of the sample space Probability model = a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events. Sample space: Some events: Rolling a 2, rolling an even #, rolling a # less than 5 How many outcomes are there if you roll 2 dice? Example: Rolling a 6-sided die Here is a chart of the possible outcomes for two dice: Each individual outcome has a probability of ________. Give the sample space for the event A = Roll a 5: Find P(A). Roll 2 3 4 5 6 7 Prob. Complete the table 8 9 10 11 12 1. Any probability is a number between 0 and 1. In symbols: For any event A, . ◦ An event with a probability of 0 never occurs (=impossible) ◦ An event with a probability of 1 occurs with every trial (=certain) ◦ An event with probability of 0.5 occurs in half the trials in the long run. Probability Rules 2. All possible outcomes together must have probability 1. In symbols: space S, P(S) = 1. Example: Check this rule by adding the probabilities on the table above: Probability Rules or for any sample 3. If two events have no outcomes in common and can never occur together ( = disjoint), the probability that one or the other event occurs is the sum of their individual probabilities. ◦ In symbols: P(A or B) = P(A) + P(B) ◦ = Addition Rule for Disjoint Events ◦ “or” is the key word meaning to add probabilities together ◦ Disjoint is also called mutually exclusive ◦ An example of disjoint events: Rolling a 2 and rolling an odd # ◦ The addition rule extends to more than 2 events as long as no two have any outcomes in common. Probability Rules 4. The probability that an event does not occur is 1 minus the probability that the event does occur. ◦ In symbols: P(A does not occur) = 1 – P(A) ◦ Called the complement of the event ◦ The probability of an event and its complement always add to 100% or 1. ◦ Example: Probability of rolling a 5 on 2 dice = ________ ◦ Probability of not rolling a 5 on 2 dice = _________________ Probability Rules Chapter 10: Introducing Probability Lesson 2 Discrete = a model with a finite sample space ◦ List in a table: All individual outcomes All individual probabilities ◦ Check table values for validity: Is 0 P(X) 1 ? Is P(x) 1 ? 1.02 is OK) (allow for round-off error = .98 to ◦ Use the table to find probabilities of events involving more than one outcome by adding necessary individual probabilities. Discrete Probability Models Faked numbers in tax returns, invoices, or expense account claims often display patterns that aren’t present in legitimate records. Some patterns, like too many round numbers, are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the first digits of numbers in legitimate records often follow a model known as Benford’s Law. Call the first digit of a randomly chosen record X for short. Benford’s Law gives this probability model for X (note the first digit cannot be 0): Example: (10.7 p 255) First Digit X Probability 1 2 3 .301 .176 .125 4 5 6 7 8 9 .097 .079 .067 .058 .051 .046 Find the probability the first digit is 1 Find the probability that the first digit is at least 6. Find the probability the first digit is greater than 6. Example cont… Worksheet... Chapter 10: Introducing Probability Lesson 3 Continuous Probability Models ◦ Continuous = a model that assigns probabilities as areas under a density curve The area under the curve above any range of values is the probability of an outcome in that range This model is used when assigning individual probabilities to outcomes is impossible because there is an infinite number of outcomes ◦ Ex: the actual ounces of water in a glass, the amount of time it takes to finish a test ◦ (both could have many decimal values) The total area under the density curve = 1 which corresponds to a total probability of 1 Probabilities are assigned to intervals of outcomes rather than individual outcomes. (individual outcomes actually have P(x)=0) We will use the Normal curve for our models—we will use the standard normal table or the TI-83 to get the probabilities Continuous Probability cont… On the TI-83: ◦ 2nd – VARS – #2:normalcdf( , For Less than: normalcdf( , greater than: normalcdf( , ) ) ) For between two values: normalcdf( , ** Round all answers properly to 4 decimal places ) Heights of young women are normally distributed with inches and inches. Find the following probabilities: ◦ A woman is between 68 and 70 inches ◦ A woman is 72 inches or taller. ◦ A woman is 60 inches or shorter Example Iowa test vocabulary scores for 7th grade students are normally distributed with and ◦ Write in terms of X: “the student chosen has a score of 10 or higher” ◦ Find the probability of the event X. Example HW page 269 #49,51,53 Chapter 10: Introducing Probability Lesson 4 Random Variables ◦ = a variable whose value is a numerical outcome of a random phenomenon ◦ A probability distribution of a random variable X tells us what values X can take and how to assign probabilities to those values. Random variables are denoted by capital letters (usually X and Y) Two types: discrete and continuous Discrete = random variables that have a finite list of possible outcomes ◦ The distribution of X assigns a probability to each of these outcomes X Ex: possible rolls for 2 standard dice X = {2,3,4,5,6,7,8,9,10,11,12} 2 3 4 5 6 7 8 9 10 11 12 P(X) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 Continuous = random variables that can take on any value in an interval with probabilities given as areas under a density curve ◦ Ex: volume of water in a 16.9 ounce bottle could be any value (decimals) depending on filling-error of machines Do page 261 #16 & 17 below: