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Probability Diagrams (or Possibility Diagrams)
When an experiment is more complex constructing a probability diagram or possibility diagram may be helpful.
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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