Page 28 - DMTH404_STATISTICS

Statistics

Notes The above assignment is also referred to as the assignment in case of equally likely outcomes.
1
You can check that in this case, the total of the probabilities of all then points is n   1. In fact,
n
this is a valid assignment even from the axiomatic pht’of view.
Now suppose that an event A contains m points. Then under the classical assignment, the
probability P(A) of A is m/n. The early probabilists called m; the number of cases favourable to
A and n, the total number of cases. Thus, according to the classical definition,

Number of cases favourable to A
P(A) 
Total number of cases
We have already mentioned that this is a valid assignment consistent with the Axiom in Sec. 6.2.
Therefore, it follows that the probabilities of events, defined in this manner, possess the properties
P1 – P7.
We now give some examyies based on this definition.

Example 5: Two identical symmetric dice are thrown. Let us find the probability of
obtaining a total score of 8.
The total number of possible outcomes is 6 × 6 = 36. There are 5 sample points, (2,6), (3,5), (4,4),
(5,3), (6,2), which are favourable to the event A of getting a total score of 8. Hence the required
probability is 5/36.

Example 6: If each card of an ordinary deck of 52 playing cards has the same probability
of being drawn, let us find the probability of drawing.
(i) a red king or a black ace
(ii) a3, 4, 5, 6 or 8?

Let’s tackle these one by one
(i) Since there are two red kings (diamond and heart) and two black aces (spade and club), the
number of favourable cases is 4. The required probability is 4/52 = 1/13.

(ii) There are 4 cards of each of the 5 denominations 3, 4, 5, 6 and 8. Thus, the total number of
favourable cases is 20 and the required probability is 20/52 = 5/13.
You must have realised by this time that in order to apply the ctassical definition of probability,
you must be able to count the number of points favourable to an event A as well as the total
number of sample points. This is not always easy. We can, however, use the theory of
permutations and combinations for this purpose.
To refresh your memory, here we give two important rules which are used in counting.
1) Multiplication Rule : If an operation is performed in n ways and for each of these n ways,
1 1
a second operation can be performed in n ways, then the two operations can be performed
2
together in n n ways. See Fig.
1 2

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