Mixed probability and sample spaces

When answering questions involving probability, it helps to have in front of us a way to represent all possible outcomes. 

We can do this by creating what's called a sample space.

A sample space can take on many forms and the most common include:

• A list
• An array
• A table
• A tree diagram
• A Venn diagram
• A two-way table

A List

Example 1:

Let's say we're interested in how many Personal Identification Numbers (PINs) can be made from the digits $1$1, $2$2, $3$3 and $8$8.

To see all options in front of us we can systematically list all combinations as follows:

$1238$1238	$2138$2138	$3128$3128	$8123$8123
$1283$1283	$2183$2183	$3182$3182	$8132$8132
$1823$1823	$2318$2318	$3218$3218	$8213$8213
$1832$1832	$2381$2381	$3281$3281	$8231$8231
$1328$1328	$2813$2813	$3812$3812	$8312$8312
$1382$1382	$2831$2831	$3821$3821	$8321$8321

At a glance we can now see there are $24$24 combinations and we can easily use our list to determine probabilities.

a)  What is the probability that a PIN starts with a $3$3 and ends with a $2$2?

Think: We look down the column where all PINs start with $3$3 and count those that also end with a $2$2.

Do: $\frac{2}{24}$224​

b)  What is the probability that a PIN starts with a $3$3 or ends with a $2$2?

Think: This time we need to count all the PINs that either start with a $3$3, end with a $2$2 OR both.

Do: $\frac{10}{24}$1024​

An Array

Example 2:

Suppose we're interested in the outcomes from rolling a normal six sided dice and spinning this spinner.

A great way to systematically do this is by creating what we can call an array (you can also simply call it a table).

	red	yellow	blue	green
1	1,red	1,yellow	1,blue	1,green
2	2,red	2,yellow	2,blue	2,green
3	3,red	3,yellow	3,blue	3,green
4	4,red	4,yellow	4,blue	4,green
5	5,red	5,yellow	5,blue	5,green
6	6,red	6,yellow	6,blue	6,green

We can now easily see that there are 24 total possible outcomes.

A Table

A table is similar to an array, but instead of listing with the use of commas what results from the two categories in the experiment, we can use a table to represent what can happen when we perform a calculation using the two categories in an experiment.

Example 3:

A six sided dice is rolled and the following spinner is spun.

The sum of the results is then recorded.

a)  Use a table to represent all possible outcomes of this experiment.

		Spinner
		1	2	3	4	5
Dice	1	2	3	4	5	6
	2	3	4	5	6	7
	3	4	5	6	7	8
	4	5	6	7	8	9
	5	6	7	8	9	10
	6	7	8	9	10	11

b)  Determine the probability that an even number was spun and a sum that is a multiple of four was obtained.

Think:  On the spinner there are two even numbers, $2$2 and $4$4. We now look down those columns for any sums that are a multiple of $4$4.

Do:  $\frac{3}{30}$330​

c)  Determine the probability the sum was greater than $7$7 or a multiple of $4$4.

Think:  We need to look in the table for all sums greater than $7$7, or that are a multiple of $4$4, or both. I've highlighted those in the table below.

		Spinner
		1	2	3	4	5
Dice	1	2	3	4	5	6
	2	3	4	5	6	7
	3	4	5	6	7	8
	4	5	6	7	8	9
	5	6	7	8	9	10
	6	7	8	9	10	11

Do:  $\frac{13}{30}$1330​

A Tree Diagram

We've discussed tree diagrams in detail before, you can review that here if you'd like to.

A Venn Diagram and a Two-Way Table

Venn diagrams and Two-way tables are two different formats for representing the same ideas. You can review those here. 

Example 4:

Sam thought that most students at her school had blonde hair and blue eyes. She surveyed 30 students at her school and found that 20 had blonde hair and 17 had blue eyes. Seven students had neither blonde hair nor blue eyes.

a)  Draw a Venn diagram to represent this situation.

We can begin by putting the information we have into a Venn diagram. Notice how I've used the rectangle in the top right and the loops in the circles to keep track of the totals.

Now we need to complete all sections in the regions of the Venn diagram. We can do so with the following calculations.

b)  Create a two-way table to represent the same information.

	Blonde	Not Blonde	Total
Blue	14	3	17
Not Blue	6	7	13
Total	20	10	30

Now check that you agree with the following probabilities and are able to use each diagram.

c)  Calculate the probability that a student did not have blonde hair.

$\frac{10}{30}$1030​

d)  Calculate the probability that a student did not have blonde hair or did not have blue eyes.

$\frac{7+3+6}{30}$7+3+630​

=$\frac{16}{30}$1630​

More Worked Examples

QUESTION 1

QUESTION 2

QUESTION 3