Let's look at a simple counting problem.
Example 1
Given a small set of numbers denoted by A = {-4, -2, 1, 3, 5, 6, 7, 8, 9, 10}, it is easy to see that there are a total of 10 numbers in A.
Now, if we are interested in determining the number of elements of the events E_{1}, E_{2}, and E, which are defined as:
- E_{1} = choosing a negative number from A
- E_{2} = choosing an odd number from A
- E = choosing a negative or an odd number from A.
Apply the Addition Principle to determine the number of outcomes of events E using E_{1} and E_{2}.
Solution
E_{1} = {-4, -2}, and the number of outcomes for E_{1} is n(E_{1}) = 2.
E_{2} = {1, 3, 5, 7, 9}, and the number of outcomes for E2 is n(E_{2}) = 5.
Events E_{1} and E_{2} are mutually exclusive because there is no common outcomes in the list of E_{1} and E_{2}. So the number of outcomes of event E is
This answer can be confirmed if we can list out all the elements of E. In this example the outcomes of E is
This gives n(E) = 7, which is the same answer as above.
Addition Principle
For a series of mutually exclusive events E_{1}, E_{2}, E_{3},. . . ,E_{n}, and each event, say E_{i} has n(E_{j}) outcomes regardless of the process made on the previous events.
Then, the total number of possible outcomes is given by
Example 2
In how many ways can a number be chosen from the set
such that
a. | it is a multiple of 3 or 8? |
b. | it is a multiple of 2 or 3? |
Solution
a. | Let | E_{1} = multiples of 3: |
E_{1} = {3, 6, 9, 12, 15, 18, 21}, so n(E_{1}) = 7. | ||
Let | E_{2} = multiples of 8: | |
E_{2} = {8, 16}, so n(E_{2}) = 2. | ||
Events E_{1} and E_{2} are mutually exclusive, so | ||
n(E) = n(E_{1}) + n(E_{2}) = 7+2 = 9 | ||
b. | Let | E_{1} = multiples of 2: |
E_{1} = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22}, so n(E_{1}) = 11. | ||
Let | E_{2} = multiples of 3: | |
E_{2} = {3, 6, 9, 12, 15, 18, 21}, so n(E_{2}) = 7. | ||
The events E_{1} and E_{2} are not mutually exclusive since they contain the same elements {6, 12, 18}. If the problem involves events that are not mutually exclusive, we can handle the counting as follows: | ||
where E1 ∩ E2 means 'the intersection of the sets E1 and E2'. |
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