FSR - Probability Distributions Lesson

Probability Distributions

Introduction

We can use theoretical probability to create a probability distribution for a certain situation.  A theoretical probability is a probability created from what we know about a situation (without conducting an experiment).  When we use theoretical probability, we often make a table of all of the possible outcomes and then analyze those possibilities.

Let's assign the random variable X to be the sum after rolling two dice.   A random variable is used for probabilities (it is different than using the letter x to represent some number!).   Let's go ahead and determine all of the possible outcomes.

	1	2	3	4	5	6
1	(1, 1)	(1, 2)	(1, 3)	(1, 4)	(1, 5)	(1, 6)
2	(2, 1)	(2, 2)	(2, 3)	(2, 4)	(2, 5)	(2, 6)
3	(3, 1)	(3, 2)	(3, 3)	(3, 4)	(3, 5)	(3, 6)
4	(4, 1)	(4, 2)	(4, 3)	(4, 4)	(4, 5)	(4, 6)
5	(5, 1)	(5, 2)	(5, 3)	(5, 4)	(5, 5)	(5, 6)
6	(6, 1)	(6, 2)	(6, 3)	(6, 4)	(6, 5)	(6, 6)

So, there are 36 different possible outcomes. Let's create a probability distribution for all of the different possibilities for X. The possible sums could be 2 to 12. 

X = sum of two rolls	Number of outcomes that satisfy that sum	Probability
2	1	$P\left(X=2\right)=\frac{1}{36}$
3	2	$P\left(X=3\right)=\frac{2}{36}=\frac{1}{18}$
4	3	$P\left(X=4\right)=\frac{3}{36}=\frac{1}{12}$
5	4	$P\left(X=5\right)=\frac{4}{36}=\frac{1}{9}$
6	5	$P\left(X=6\right)=\frac{5}{36}$
7	6	$P\left(X=7\right)=\frac{6}{36}=\frac{1}{6}$
8	5	$P\left(X=8\right)=\frac{5}{36}$
9	4	$P\left(X=9\right)=\frac{4}{36}=\frac{1}{9}$
10	3	$P\left(X=10\right)=\frac{3}{36}=\frac{1}{12}$
11	2	$P\left(X=11\right)=\frac{2}{36}=\frac{1}{18}$
12	1	$P\left(X=12\right)=\frac{1}{36}$

Note: All of these probabilities add to 1!

We can graph this probability distribution using a histogram, we'd put each outcome on the x-axis and the probability on the y-axis.

 

 

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Probability Distribution Practice

 

IMAGES CREATED BY GAVS