Addition Rule of Probability

The general formula for the probability of either one of two events, A or B, happening at all is:

If the events A and B are mutually exclusive (that is, cannot happen at the same time), the Addition Rule is simplied to the following:

Example 1  (with mutually exclusive events)

Position Number Only Child of the marriage 49 Oldest Child of the marriage 35 Youngest Child of the marriage 72 Adopted Child from another family 15 Step Child 19 Foster Child 10

A study was made of 200 children who exhibited evidence of abuse by an adult. The table on the left shows each abused child's position in the family. What is the probability of randomly selecting a child from the group of children described above and finding that the child was either a step child or an adopted child? What is the probability of randomly selecting a child from the group of children described above and finding that the child was either a step child or an adopted child or a foster child? What is the probability of randomly selecting a child from the group of children described above and finding that the child was neither a step child nor a foster child nor an adopted child?

Analysis of the Problem

The sample space S of this experiment is based upon the set consisting of the entire group of abused children. The "experiment" consists of categorizing each child of the group into his/her respective position in the family. The family position of a child is the "outcome" of the experiment for that particular child. (In this case, no child could fall into more than one category.) Therefore, the sample space S is actually the set of family positions, one for each child in that group.

The total # of outcomes = sample size = n = 200. (one outcome for each of the 200 children)
Let A be the event that a step child is chosen from the group of children. The event A consists of the outcome: family position is "step child".
The # of A-outcomes = 19.
Let B be the event that an adopted child is chosen from the group of children. The event B consists of the outcome: family position is "adopted child".
The # of B-outcomes = 15.
P(A) = # of A-outcomes / n = 19 / 200 = 0.095
P(B) = # of B-outcomes / n = 15 / 200 = 0.075
Observe that A and B are mutually exclusive events.

Solution to 1.a

The Addition Rule gives us a way of calculating the probability that the child was either a step child or an adopted child as follows:

The probability that a randomly selected child is either a step child or an adopted child is 0.17.

Solution to 1.b

From part 1, we know that P(A or B) = 0.17.
Let C be the event that a foster child is chosen from the group of children.
The # of C-outcomes = 10.
P(C) = # of C-outcomes / n = 10 / 200 = 0.05
Observe that A, B, and C are mutually exclusive events. The children are slotted into only one family-postion category. The intersection of any two of the categories is empty. Using the Addition Rule again, we get:

The probability that a randomly selected child is either a step child or an adopted child or a foster child is 0.22.

Solution to 1.c

The probability that the child was neither a step child nor a foster child nor an adopted child may be obtained by adding up the probabilities of all the events in which a child was chosen outside of the above-stated three categories. The Addition Rule may be applied in a way similar to part 2.

However, there is another way to think about this. All the events in which a child was chosen outside of the above-stated three categories together constitute the complement of the combined A, B, and C categories. From the Complement Rule, we can obtain the probability we want. From part 2, we know that P(A or B or C) = 0.22.
Therefore, the probability that the child was neither a step child nor a foster child nor an adopted child is

The probability that a randomly selected child is neither a step child nor a foster child nor an adopted child is 0.78.

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Example 2

Last Updated by SAS October 07, 2026, 16:00:54