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.

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.

whatever you can spend with $11.53 or under.

step-by-step explanation:

20 - 8.47 = 11.53. take away the money he used for lunch from the starting amount of 20.

i found two solutions,   n = 2 • ± √3 = ± 3.4641

step-by-step explanation: