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Statistics
Chapter 3: Introduction to Discrete Random Variables
What are 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, 1.95434, 2.7764, 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.
X
0
1
2
3
P(x)
.1
.2
.3
.4
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
𝑃 𝑥 = 𝑎 = 𝑛𝐶𝑎 ∗ 𝑝𝑎 ∗ 𝑞 𝑛−𝑎
A is the number of successes from the
trials
P is the probability of the event
occurring
Q is the probability of the event not
occurring
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
 Number of desired outcomes divided by
the total number of outcomes
𝑛𝑢𝑚𝑏𝑒𝑟 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
 P(desired)=
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Experimental Probability
 Number of times the desired outcome occurs
divided by the total number of trials.
 𝑃 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑐𝑐𝑢𝑟𝑠
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠
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?