Page 101 - DMTH404_STATISTICS

Unit 7: Modern Approach to Probability

(i) Let A be the event that the sum of spots is 5 and B be the event that their sum is 10. Thus, Notes
we can write
A = {(1, 4), (2, 3), (3, 2), (4, 1)} and B = {(4, 6), (5, 5), (6, 4)}
We note that A B b g   , i.e. A and B are mutually exclusive.

4 3 7
 By addition theorem, we have (A  ) B  P A ( ) +  .
P
( ) P B+
36 36 36
(ii) Let C be the event that there is a doublet and D be the event that the sum is less than 6.
Thus, we can write
C = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} and
D = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}
Further, C D b g = {(1, 1), (2, 2)}

6 10 2 7
By addition theorem, we have (C  D )  +   .
P
36 36 36 18
Alternative Methods:
It is given that n(A) = 4, n(B) = 3 and n(S) = 36. Also n A B b
(i) g  0 . Thus, the corresponding
nine-square table can be written as follows:

B B Total
A 0 4 4
A 3 29 32
Total 3 33 36

29 7
From the above table, we have (A  B ) 1   .
P
36 36
Here n(C) = 6, n(D) = 10, n C D b
(ii) g  2 and n(S) = 36. Thus, we have
C C Total
D 2 8 10
D 4 22 26
Total 6 30 36

22 7
Thus, (C  D ) 1 P C  (  D ) 1   .
P
36 18
Example 26: Two unbiased coins are tossed. Let A be the event that the first coin shows
1
a tail and A be the event that the second coin shows a head. Are A and A mutually exclusive?
2 1 b 1 b 1 2
A
A
Obtain P A  g and P A  g . Further, let A be the event that both coins show heads and
1 b
2
2
1
A
A be the event that both show tails. Are A and A mutually exclusive? Find P A  g and
2 1 b 1 2 2
A
P A  g .
2
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