Addition Rule of Probability
Example 3
A jelly bean manufacturer did of survey of its customers as to their favourite colour of jelly bean. The results of the survey indicated that: P(orange) = 0.32, P(lemon yellow) = 0.37, and P(black) = 0.11.
- What is the probability that a randomly selected customer would choose either the colour orange or the colour black?
- What is the probability that a randomly selected customer would choose either orange, lemon yellow or black?
- What is the probability that a randomly selected customer would not choose any of those colours?
Analysis of the Problem
By the wording of the question asking for the probability of either the colour orange or the colour black, you should immediately think of the Addition Rule. The next step is to analyse the problem into what is the statistical experiment, the sample space, the sample size, the two events, and their probabilities so that all the pieces of information can be placed into the Addition Rule to solve the problem.
The sample space S of this experiment is based upon the set consisting of the group of customers of the jelly bean manufacturer. This statistical experiment consists of asking a customer to choose his/her favourite colour of jelly bean. Since each customer polled was asked his/her favourite, these categories are mutually exclusive because only one colour could be chosen at a time.
The favourite colour chosen is an "outcome" of the eperiment for that particular customer. The sample space S is, therefore, a list (set) of the colours the customers have chosen, one for each customer polled.
As the total number of customers polled is not given in the question, we are not able to give the sample size. Neither are we given how many customers chose a given colour of jelly bean. What we are given are the probabilites for a select few colours.
Now we proceed with the solution by listing all the probabilities given in the problem.
Let A be the event that the customer chooses the colour orange. The event A consists of the outcome: favourite colour is "orange".
Let B be the event that the customer chooses the colour lemon yellow. The event B consists of the outcome: favourite colour is "lemon yellow".
Let C be the event that the customer chooses the colour black. The event B consists of the outcome: favourite colour is "black".
P(A) = 0.32, P(B) = 0.37, P(C) = 0.11
P(A and B) = 0, P(A and C) = 0, P(B and C) = 0
Now the probability of a randomly selected customer choosing either orange or black, therefore, is P(A or C).
We substitute into the general form of the Addition Rule to get:
The probability that a randomly selected customer would choose either orange or black is 0.43.
Solution to 3.b
The probability that a randomly selected customer would choose either orange, lemon yellow, or black also involves the Addition Rule, but there are 3 events instead of 2. However, we have already calculated the probability of either orange or black — so we could treat it as one event. Then we treat the choosing of lemon yellow as the second event.
Therefore, substituting into the general form of the Addition Rule, we get:
Observe that P(A or C) = .43 from part a, P( (A or C) and B ) = 0 since all the joint probabilities for mutually exclusive events are 0.
The probability that a randomly selected customer would choose either orange, lemon yellow, or black is 0.8.
Solution to 3.c
The probability that a randomly selected customer would not choose any of the stated colours involves the Complement Rule since a customer not choosing orange, lemon yellow, or black is the complement of choosing orange, lemon yellow, or black.
We substitute into the general form of the Complement Rule to get:
The probability that a randomly selected customer would not choose any of the stated colours is 0.2.
Example 1
Last Updated by SAS October 07, 2026, 16:00:54
