Objective
In this lesson, you will learn about probability distributions and how they describe the probabilities associated with different possible values of a random variable.
Concept
Assume you have an unfair coin that is weighted to land on heads 65% of the time. If you flip that coin 3 times and let represent the number of tails you get, what is the probability distribution for ?
Look to the end of the lesson for the answer.
Watch This
http://youtu.be/s2S1oD3ovps statslectures  Discrete Probability Distributions
Guidance
A probability distribution is a list of each value a random variable can attain, along with the probability of attaining each value. In other words, the probability distribution of an event is sort of a map of how each possible outcome relates to the chance it will happen.
For instance, the probability distribution of flipping a coin twice is:
heads, heads = 25%, heads, tails = 25%, tails, heads = 25%, and tails, tails = 25%.
If we define the random variable to be the number of heads you get when you flip a coin twice, we could create the following probability distribution table for :







There are various ways of visualizing a probability distribution, and we will review that concept in another lesson. For now, we focus on identifying what a probability distribution is, and how to calculate it for a particular event.
Example A
In Chi’s class, 4 students have one parent, 7 have two parents, and 1 student lives with his uncle. Let be the number of parents of a randomly selected student from the class. Create a probability distribution for .
Solution:
Set random variable to be the number of parents:
Now find the probability of each , noting that there are 12 students total:
Example B
Roll two fair sixsided dice. Let equal the sum of the dice. Create a probability distribution for .
Solution: Make a list of the individual probabilities of each of the 36 possible outcomes:
Example C
Janie wants to evaluate the probabilities of pulling various cards from a deck. She sets the discrete random variable to be the number of diamonds she gets over the course of three trials, if each trial consists of pulling, recording, and replacing one random card from a standard deck. What is the probability distribution of ?
Solution:
To evaluate the probability distribution of
, Janie needs to identify the probability of each of the possible values of
. Note that the chance she will pull a diamond is
or
, meaning that the chance she will
not
pull a diamond is
:

For
, the total probability is:
(see the three possible outcomes resulting in
below)
 Diamond, other, other :
 Other, Diamond, other :
 Other, other, Diamond :

For
, the total probability is:
 Diamond, Diamond, other :
 Diamond, other, Diamond :
 Other, Diamond, Diamond :

For
, the probability is :
 Diamond, Diamond, Diamond:
Concept Problem Revisited
Assume you have an unfair coin that is weighted to land on heads 65% of the time. If you flip that coin 3 times and let represent the number of tails you get, what is the probability distribution for ?
If each throw has a 65% chance of heads, then it has a 35% chance of tails:
 For , we could have THH, HTH, or HHT. Each of those has a chance of occurring, so
 For , we could have TTH, THT, or HTH. Each has a chance, so
 For , we could have only TTT, with a chance of
Vocabulary
A probability distribution is a list of each value a random variable can attain, along with the probability of attaining each value.
Guided Practice
 Create a probability distribution for number of heads when you flip a coin 3 times.
 Let be the number of chocolate chip cookies you get if you randomly pull and replace two cookies from a jar containing 6 chocolate chip, 4 peanut butter, 8 snickerdoodle, and 12 sugar cookies. Create a probability distribution for .
 Let be the score of a single student chosen at random from Mr. Spence’s class. Create a probability distribution for , given the following:
Number of Students 
Test

11 
87 
7 
89 
13 
92 
9 
94 
6 
96 
Solutions:
1. Write out all the possibilities:
2. There are a total of 30 cookies, the probability of pulling a chocolate chip cookie is , so the probability of not pulling a chocolate chip is
 For we have to pull a nonchocolate chip both times:
 For we could either pull the chocolate chip cookie first or second, so we get
 For we have to pull chocolate chip both times, so we have
3. There are a total of 46 students in Mr. Spence’s class, so there are 46 scores. The probability of a random student having score is the same as that score’s portion of the total number of scores:
Practice
1. What is a probability distribution?
2. What is a random variable?
3. What is the difference between a discrete and a continuous random variable?
For problems 47, refer to the following table:
2 
3 
4 
5 
6 
7 
8 
9 
10 

.04 
.12 
.16 
.16 
.12 
.04 
4. Assuming the table is a probability distribution for discrete random variable , which is the sum of two dice rolled once, how many sides does each die have?
5. What is ?
6. What is ?
7. What is ?
8. Roll two sevensided dice once. Let be the sum of the two dice. Create a probability distribution for .
9. Flip a fair coin 3 times, let be the number of heads. Create a probability distribution for .
10. Let be the sum of two standard fair dice. Create a probability distribution for , if the experiment consists of a single roll of both dice.