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yoboi33
13.03.2020 •
Mathematics
Let A and B be events having a positive probability. State whether each of the following statements is (i) necessarily true, (ii) necessarily false, or (iii) possibly true.
a. If A and B are mutually exclusive, then they are independent.
b. If A and B are independent, then they are mutually exclusive.
c. P(A) = P(B) = 0.6, and A and B are mutually exclusive.
d. P(A) = P(B) = 0.6, and A and B are independent.
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Ответ:
A) It's necessarily false
B) It's necessarily false
C) It's necessarily false
D) It's possibly true.
Step-by-step explanation:
From the question, probability of events A and B is positive.
Thus;
P(A) > 0 and P(B) > 0
A) If the two events are mutually exclusive, then; P(A ∩ B) = 0 eq(1)
If the two events are independent, then;
P(A) x P(B) = P(A ∩ B) - - - - - eq(2)
Since the events are mutually exclusive, then;
P(A) x P(B) > 0 - - - - - eq(3)
Combining eq 2 and 1,we have
P(A) x P(B) = 0 - - - - eq(4)
Since equation 3 and 4 are contradictory, thus we can say that the events are not independent. And so, it is necessarily false.
B) if events are independent, then
P(A) x P(B) = P(A ∩ B)
And P(A) x P(B) > 0
Thus, it implies that;
P(A ∩ B) > 0 - - - - eq(5)
For the events to be mutually exclusive, P(A ∩ B) = 0 - - - - (6)
But eq 5 and 6 are contradictory, and thus events are not mutually exclusive and so it's necessarily false.
C) P(A) = P(B) = 0.6
P(A) + P(B) = 0.6 + 0.6 = 1.2
From union of events;
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
And every probability is less than 1.
Thus, P(A ∪ B) < 1
Thus,
P(A) + P(B) - P(A ∩ B) < 1
So rearranging, we have;
P(A) + P(B) - P(A ∪ B) < 1
P(A) + P(B) - 1 < P(A ∪ B)
0.6 + 0.6 - 1 < P(A ∪ B)
So P(A ∪ B) > 0.2
So events are not mutually exclusive and so it's necessarily false
D) we have seen that;
P(A ∪ B) > 0.2
And that ; P(A) = P(B) = 0.6
Thus,
P(A) x P(B) = 0.6 x 0.6 = 0.36
Since, P(A ∪ B) is greater than 2,it is possible that it may be equal to P(A ∪ B). So it's possibly true.
Ответ:
i'd say you'll have to change the numbers to variables and vice versa.hope it ,happy homework/ study/ exam!