Probability Diagrams (or Possibility Diagrams)

A possibility diagram (also called a sample space diagram) is a visual tool used in probability to systematically list all the possible outcomes of two (or sometimes more) independent events. It’s particularly useful when each outcome is equally likely.

What is a Sample Space?
In probability, the sample space is the set of all possible outcomes of a random experiment. For example:
Flipping a coin: Sample Space = {Heads, Tails} or {H, T}
Rolling a single die: Sample Space = {1, 2, 3, 4, 5, 6}

When an experiment is more complex constructing a probability diagram or possibility diagram may be helpful in constructing the sample space.

Example:
The diagram shows two spinners, each of which is divided into 4 equal sectors. Each spinner has a pointer which, when spun, is equally likely to come to rest in any of the four equal sectors.

Probability Worksheets
Practice your skills with the following worksheets:
Printable & Online Probability Worksheets

In a game, each pointer is spun once.
Find the probability that
a) the pointers will stop at the same number
b) the first spinner shows the larger number.

Solution:
Construct the probability diagram. Each dot represents a possible outcome according to the coordinates.

a) Let A = event of getting the same number on the two spinners.
From the probability diagram, n(A) = 4, n(S) = 16
\(P(A) = \frac{n(A)}{n(S)} = \frac{4}{16} = \frac{1}{4}\)

b) Let B = event the first spinner shows the bigger number.
From the probability diagram, n(B) = 6
\(P(B) = \frac{n(B)}{n(S)} = \frac{6}{16} = \frac{3}{8}\)

Example:
Two fair dice are thrown together. Find the probability that the sum of the resulting number is
a) odd
b) a prime number

Solution:
Construct the following probability diagram showing the sums:

a) Let A be the event that the sum is odd
From the probability diagram, n(A) = 18
\(P(A) = \frac{18}{36} = \frac{1}{2}\)

b) Let B be the event that the sum is a prime
Count the number of 2, 3, 5, 7 and 11 in the probability diagram.
n(B) = 15
\(P(B) = \frac{15}{36} = \frac{5}{12}\)

Example:
X = {1, 2, 3} and Y = {4, 5, 6}. An element x is selected from X and an element y is selected from Y.
Complete the following probability diagrams for x + y and x × y

a) Find the probability that the sum x + y is:
i) prime
ii) greater than 7
b) Find the probability that the product xy is:
i) odd
ii) at most 10

Solution:
a) The following probability diagrams for x + y and x × y


a) The probability that the sum x + y is:
i) prime
Let S be the sample space, and A be the event that the sum is prime.
From the probability diagram, n(A) = 4 ; n(S) = 9
\(P(A) = \frac{4}{9} \)

ii) greater than 7
Let S be the sample space, and B be the event that the sum is greater than 7.
From the probability diagram, n(B) = 3 ; n(S) = 9
\(P(B) = \frac{3}{9} = \frac{1}{3}\)

b) The probability that the product xy is:
i) odd
Let S be the sample space, and C be the event that the product is odd.
From the probability diagram, n(C) = 2 ; n(S) = 9
\(P(C) = \frac{2}{9} \)

ii) at most 10
Let S be the sample space, and D be the event that the product is at most 10.
From the probability diagram, n(D) = 5 ; n(S) = 9
\(P(D) = \frac{5}{9} \)

Probability and Possibility Space diagrams
Examples:

1. Two normal 6-sided fair dice are thrown and the total score is recorded. Construct a possibility space digram showing all possible outcomes and find the probability of scoring a total of 7.
   
   
2. A normal 6-sided fair dice is thrown and a coin is tossed. Draw a possibility space diagram showing all the outcomes and find the probability of getting a number less than 3 and a head.

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How to use diagrams to help solve probability problems?
Example:
What is the probability of getting a certain number roll in Monopoly?

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How to use a possibility space diagram or sample space diagram to solve word problems?
Example:
Two dice are thrown and their product noted. Draw a possibility space diagram to show all the possible outcomes. Use your diagram to find:
(a) the probability of the product being 12.
(b) the probability that the product is less than 6.
(c) the probability that the product is 20 or more.

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