Suppose that two independent events E_{1} and E_{1} can result in n(E_{1}) and n(E_{2}) possible outcomes, respectively, and that there are no restrictions on choice.
The total number of different combinations of possible outcomes from E1 and E1 can then be determined by using the Multiplication Principle, that is
Multiplication Principle
If a process can be performed in a series of independent successive events E_{1}, E_{2}, E_{3},. . . , E_{n} and each event, say E_{i}, can be completed in n(E_{j}) ways regardless of the process made on the previous events.
Then the total number of different combinations of completing the entire process is given by
Example 3
What is the total number of possible outcomes when you flip a fair coin twice?
Solution
Let T be the 'tail' and H be the 'head' of a fair coin. The events are:
- E_{1} = first flip (2 possible outcomes: H or T, so n(E_{1})= 2)
- E_{2} = second flip (2 outcomes: H or T, so n(E_{2}) = 2)
The events are independent and that the flip does not affect the outcome of the other flip.
Hence the total number of possible outcomes is
[We could also list the outcomes as: HH HT TH TT].
In the previous example, you can also model the 4 possible outcomes using a tree diagram or an outcome table, as shown below:
1st Flip |
2nd Flip |
Outcomes |
---|---|---|
H |
H |
HH |
H |
T |
HT |
T |
H |
TH |
T |
T |
TT |
Outcome Table |
Example 4
The life insurance policies of an insurance company are classified by:
- age of the insured: under 25 years, between 25 years and 50 years, over 50 years old;
- sex: M or F; and
- marital status: single or married.
What is the total number of classifications?
Solution
The events are:
E_{1} = age of the insured: | 3 age divisions, so n(E_{1}) = 3 |
E_{2} = sex: |
2 possibilities of sex, so n(E_{2}) = 2 |
E_{3} = marital status: |
2 possibilities of martial status, so n(E_{3}) = 2 |
All events E_{1}, E_{2} and E_{3} are independent, so the total number of different classifications is
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