Probability Diagrams (or Possibility Diagrams)

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.

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) =

b) Let B = event the first spinner shows the bigger number.

From the probability diagram, n(B) = 6

P(B) =

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) =

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) =

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) =

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) =

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) =

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) =

The following video shows another example of using diagrams to help solve probability problems.

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