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Statistics Chapter 3: Introduction to Discrete Random Variables

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What are variables?

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Random Variables are quantities that take on different values depending on chance or probability. Can you give some other examples? Number of cars that are Fords Discrete Random Variables represent the number of distinct values that can be counted of an event. Can you give some other examples? A random bag of jelly beans has 15 of 250 green jelly beans

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IMPORTANT NOTE Values of Discrete Random Variables are not mutually exclusive… They cannot be the same thing at the same time Example: you cannot have a car that is both a Ford and a Chevy

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Discrete Versus Continuous Discrete variables: variables which can assume only countable values Counting numbers 1,2,3,4,5,6,7,8,9,10, etc. Continuous variables: variables which can assume a countless number of values Counting numbers with decimals 1.2, 1.95434, 2.7764, etc.

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Discrete versus Continuous Discrete examples: Number of students in a class Number of cars at your house Continuous examples Person’s height Dog’s weight What other examples can you think of?

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Quantitative versus Qualitative Quantitative variables: variables that can be measured, numbers Length, height, area, volume, weight, speed, time, temperature, humidity, sound level, cost, ages, etc. Qualitative variables: variables that can be observed, observations Colors, textures, smells, tastes, appearance, beauty, etc.

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Describe quantitatively and qualitatively.

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The Probability Distribution

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Probability Distribution A probability distribution is a table, graph, or chart that shows you all the possible values of your variable and the probability associated with each of these values. NOTE: all the probabilities (percentages) must add up to equal 1

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Probability Distribution Example

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Probability Distribution Create a probability distribution table for tossing 2 coins, showing the probability of the coin landing on tails.

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Histogram A histogram is a probability distribution graph that uses horizontal or vertical bars to display data.

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Histogram Create a histogram to show the previous probability distribution question: Create a probability distribution table for tossing 2 coins, showing the probability of the coin landing on tails.

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Histogram Determine whether or not the data represents a probability distribution and create a histogram for the data. X0123 P(x).1.2.3.4

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Shuffle Board Activity Rules: There will be three teams, each with two ‘pucks’. Standard rules apply for regular shuffle board Player with the ‘puck’ the closest to the edge of the board receives all points for pucks. Other teams do not receive any points. Rotate as to which team will go first.

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Shuffle board activity Create two probability distribution charts showing the probabilities for this given game, noting the following: Probability of scoring each of the given number of points Probability of having your puck in each of the given scoring sections

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Probability Distribution Charts

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Skewed Distributions

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A Glimpse at Binomial and Multinomial Distributions

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Complements Example The probability of scoring above a 75% on a math test is 40%. What is the probability of scoring below a 75%?

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Probability of X successes in n trials

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Example A fair die is rolled 10 times. Let X be the number of rolls in which we see a 2. What is the probability of seeing a 2 in any one of the rolls? What is the probability of seeing a 2 exactly once in the 10 rolls?

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Example A fair die is rolled 15 times. Let X be the number of rolls in which we see a 2. What is the probability of seeing a 2 in any one of the rolls? What is the probability of seeing a 2 exactly twice in the 15 rolls?

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Example A fair die is rolled 10 times. Let X be the number of rolls in which we see at least one 2. What is the probability of seeing at least one 2 in any one roll of the pair of dice? What is the probability that in exactly half of the 10 rolls, we see at least one 2?

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Binomial Distribution Only 2 outcomes (true or not true) Fixed number of trials Each trial is independent of other trials

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Multinomial Distribution Multiple outcomes

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Example You are given a bag of marbles. Inside that bag are 5 red marbles, 4 white marbles, and 3 blue marbles. Calculate the probability that with 6 trials you choose 3 marbles that are red, 1 marble that is white, and 2 marbles that are blue. Replacing each marble after it is chosen.

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Using Technology to Find Probability Distributions

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To Find To do a coin toss, spin a spinner, roll dice, pick marbles from a bag, or draw cards from a deck Push APPS Go down to PROB SIM Push enter Choose which simulation you want

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Example You are spinning a spinner 20 times. How many times does it land on blue?

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Theoretical Probability

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Experimental Probability

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Example You are spinning a spinner 50 times. How many times does it land on 4? What is the theoretical probability? What is the experimental probability?

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Example A fair coin is tossed 50 times. What is the theoretical probability and the experimental probability of tossing tails on the fair coin?

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