Page 29 - DMTH404_STATISTICS

Unit 2: Methods of Enumeration of Probability

Notes
Figure 2.1

2) Addition Rule : Suppose an operation can be performed in n, ways and a second operation
can be performed in n ways. Suppose, further that it is not possible to perform both
2
together. Then the number of ways in which we can perform the first or the second
operation in n + n . See Fig. 4.
1 2
Figure 2.2

We now illustrate the use of this theory in calculating probabilities by considering some examples.
We assume that all outcomes in each of these examples are equally likely. Under this assumption,
the classical definition of probability is applicable.

Example 7: We first select a digit out of the ten digits, 0, 1,2,3, ..., 9. Then we select another
digit out of the renlaining nine. What will be the probability that both these digits are odd?
We can select the first digit in 10 ways and for each of these ways we can select the second digit
in 9 ways. Therefore, the total number of points in the sample space is 10 × 9 = 90. The first digit,
can be odd in 5 ways ( 1,3,5,7.9). and then the second digit can be odd in 4 ways. Thus, the total
number of ways in which both the digits can be odd is 5 × 4 = 20. The required probability is
20 2
therefore  .
90 9
Remark 2 : In Example 7, every digit had the same chance of being selected. This is sometimes
expressed by saying that the digits were selected at random (with equal probability).
Sele~tion~raatn dom is generally taken to be synonymous with the assignment of the same
probability to all the sample points, unless stated otherwise.

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