Answer to Question #126777 in Statistics and Probability for jaya

Question

01) a) State, with evidence, whether each of the following statements are true or false:
I. If A and B are independent, then A’ and B’ must be independent.
II. The probability of the union of two events cannot be less than the sum of their
individual probabilities.
III. The probability of the intersection of two events cannot be greater than either of their
individual probabilities.
IV. An event and its complement are not mutually exclusive.
V. If A and B are two events, the probability of A, given B, is the same as the probability of
B, given A, if the probability of A is the same as the probability of B.

Answer 1

I) P(A^cnB^c)

=1−P(A∪B)

=1−P(A)−P(B)+P(A∩B)

=1−P(A)−P(B)+P(A)P(B)

=(1−P(A))(1−P(B))

=P(A^c)P(B^c)- True

II)Using the axiom of probability, P(AUB)$\leq$ P(A)+P(B)

So it can be less than the sum of two probabilities.- False

III)The event that both A and B occur is the intersection of the events A occurs and B occurs. As such, it is a subset of each and cannot, therefore, have a larger probability than either one individually.- True

IV) The event A and its complement A’ are mutually exclusive and exhaustive because the two events cannot occur at the same time- False

V)P(A/B)= P(A and B)/ P(B) and P(B/A)= P(A and B)/ P(A)

= P(A and B)/P(B)[since P(A) is same as P(B)]

=P(A/B)

True