Page 16 - DMTH404_STATISTICS

Statistics

Notes
Examples 3: In many games of chance, a small’cube (or die) with equal sides, bearing
numbers 1, 2, 3, 4, 5, 6, or dots 1-6 on its six faces (Fig. 1.2), is used. When such a symmetric , die
is thrown, one of its six faces would be uppermost. The number (or number of dots) on the
uppermost faces is called the score obtained on the throw or roll of a die. The appropriate
sample space for the experiment of throwing a die is then R = {1, 2, 3, 4, 5, 6}.

Figure 1.2

Let A be the event that the score exceeds three and A be the event that the score is even.
1 2
Then
A = {4, 5, 6}, A = {2, 4, 6}
1 2
Therefore, A A = {4, 6} and
1 2
A A = {2, 4, 5, 6}.
1 2
Suppose now that the score is 6. We can say that A has occurred. But then A has also occurred.
1 2
In other words, both A and A have occurred. Thus, the simultaneous occurrence of A and A
1 2 1 2
corresponds to the occurrence of the event A A .
1 2
When the outcome is 5, A has occurred but A has not occurred. Further, when the outcome is 2,
1 2
A has occurred and A has not. When the outcome is 4, both A and A have occurred. In case of
2 1 1 2
each of these outcomes, 2,5 or 4, we notice that at least one of A and A has occurred. Note,
1 2
further, that A A has also occurred. Thus, the occurrence of at least one of the two events
1 2
A and A corresponds to the occurrence of A A .
1 2 1 2
Examples 4: Suppose the die in Example 3 is thrown twice. Then  is the set { (x, y)|x, y
= 1, 2, 3, . . . , 6 } consisting of thirty-six points (x, y), where x is the score obtained on the first
throw and y, that obtained on the second throw. If B is the event that the score on the first throw
1
is six and B the event that the sum of the two scores is at least 11, then
2
B = { (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }
1
and
B = { (5, 6), (6, 5), (6, 6) }.
2
What are B B and B B ? You can check that
1 2 1 2
B B = { (6, 5), (6, 6) }
1 2
and

B B = { (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }.
1 2

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