## Friday, April 27, 2012

### PROBABILITIES INVOLVING MORE THAN ONE EVENT

Probabilities involving more than one event may be calculated using the Addition Rule and Multiplication Rule. Those are as follows:

If there are several possible events, but only one could occur at any one time, the probability that one of such events will actually occur at a given time is the total of the probabilities of each individual event happening at that time.

MULTIPLICATION RULE:
If several events are independent (that is occurrence of any one has no effect on the probability of the occurrence of the others), the probability that all of the events will happen is represented by multiplying the probabilities of the individual events together.

To Illustrate: Using the following portion of a mortality table, calculate the probability that a person age 40 will die either at age 60 or at age 61.

 Age (x) lx dx 40 97353 198 41 97155 216 . . . . . . . . . 60 84650 1325 61 83325 1406 62 81919 1491

Solution:
Since only one of the events can occur, the Addition Rule that two individual probabilities are added is applicable. The probability that a person age 40 will die during the year he or she is age 60 is equal to the number of people dying at age 60 divided by the number living at age 40:

Basic equation:

Substitute the values from the given table:

(Probability of Dying at Age 60)  =  1325/97353
(Probability of Dying at Age 60) = 0.0136
Similarly, the probability that a person age 40 will die during the year he or she is age 61 is

Basic equation:

Substitute the values from the table

(Probability of Dying at age 61) = 1406/97353

(Probability of Dying at age 61) = 0.0144

The desired probability equals the total of the two individual probabilities:

The equation is:

Hence, 0.0280 is the probability that a person age 40 will die either at age 60 or 61.

To Illustrate: Using the following portions of a mortality table, calculate the probability that a mother age 55, and her daughter, age 25 will both live at least 15 more years.

 Age (x) lx 24 9953 25 9947 26 9940 . . . . . 39 9827 40 9814 41 9801 . . . . . . 54 9444 55 9388 56 9327 . . . . . . 69 7885 70 7717 71 7539

Solution:
Assuming that one event has no effect on the other, the Multiplication Rule, that two individual probabilities are multiplied is applicable. For mother;

Basic equation:

For Daughter:

Basic equation

The desired probability that both will live at least 15 years equal the two individual probabilities multiplied together:

The equation is:

The result shows that the probability that a mother, age 55, and her daughter, age 25, will both live at least 15 more years is 0.8110.