Statistics Chapter 3: Introduction to Discrete Random Variables

What are variables?

 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

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

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, , , etc.

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?

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.

Describe quantitatively and qualitatively.

The Probability Distribution

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

Probability Distribution Example

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

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

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.

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

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.

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

Probability Distribution Charts

Skewed Distributions

A Glimpse at Binomial and Multinomial Distributions

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

Probability of X successes in n trials

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?

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?

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?

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

Multinomial Distribution  Multiple outcomes

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.

Using Technology to Find Probability Distributions

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

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

Theoretical Probability

Experimental Probability

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?

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