Question

Let A and B represent events such that P(A) = 0.6, P(B) = 0.4, and P(A ∪ B) = 0.76. Compute: (a) P(A ∩ B) (b) P(Ac ∪ B) (c) P(A ∩ Bc ) (d) Are events A and B mutually exclusive? Are they independent? Explain by citing the definitions of mutual exclusivity and independence.

Answer #1

a) P(A B) = P(A) + P(B) - P(A U B)

= 0.6 + 0.4 - 0.76

= **0.24**

A Venn Diagram can be used to answer the remaining parts of the question

b) P(A^{c} U B) = 0.24 + 0.16 + 0.24

= **0.64**

c) P(A
B^{c}) = **0.36**

d) A and B are mutually exclusive if P(A B) = 0

Here, P(A B) 0

Hence, **A and B are not mutually exclusive.**

A and B are independent if P(A) x P(B) = P(A B)

P(A) x P(B) = 0.6 x 0.4 = 0.24

P(A B) = 0.24

P(A) x P(B) = P(A B)

Therefore, **A and B are independent.**

Let A and B be two events such that P(A) = 0.8, P(B) = 0.6 and
P(A B) = 0.4. Which statement is correct?
a.
None of these statements are correct.
b.
Events A and B are independent.
c.
Events A and B are mutually exclusive (disjoint).
d.
Events A and B are both mutually exclusive and independent.
e.
Events A and B are the entire sample space.

Let A and B be events with P(A) = 0.7, P(B) = 0.9, and P
(A and B) = 0.6.
Compute P (A or B)
Are A and B mutually exclusive? Explain.

Let A and B be events with P (A)= 0.3 and P (B)=0.7, and P (A or
B)=0.9. (a) Compute . (b) Are and mutually exclusive? Explain. (c)
Are and independent? Explain.

Let A and B be independent events of some sample space. Using
the definition of independence P(AB) = P(A)P(B), prove that the
following events are also independent:
(a) A and Bc
(b) Ac and B
(c) Ac and Bc

Let P[A] = P[B] = 1/3 and P[A ∩ B] = 1/10.
Find the following:
(a) P[A ∪ Bc ]
(b) P[Ac ∩ B]
(c) P[Ac ∪ Bc ]
Now, let P[A] = 0.38 and P[A ∪ B] = 0.84.
(d) For what value of P[B] are A and B mutually exclusive?
(e) For what value of P[B] are A and B independent?

Consider the following scenario:
• Let P(C) = 0.7
• Let P(D) = 0.4
• Let P(C|D) = 0.8
Q1. P(C AND D) =
Q2. Are C and D Mutually Exclusive?
Q3 Are C and D independent events?
Q4. P(D|C) =
Round your answer to two decimal places.

For each situation, determine if events A, B are independent.
Explain.
P(A | B) = 0.4,
P(B) = 0.8, and P(A) =
0.5
P(A | B) = 0.3,
P(B) = 0.8, and P(A) =
0.3
P(A) = 0.2, P(B) =
0.2, and A and B are mutually
exclusive

Consider events A, B, and C, with P(A) > P(B) > P(C) >
0. Events A and B are mutually exclusive and collectively
exhaustive. Events A and C are independent.
(a) Can events C and B be mutually exclusive? Explain your
reasoning. (Hint: You might find it helpful to draw a Venn
diagram.)
(b) Are events B and C independent?
Explain your reasoning.

Let P(A) = 0.4, P(A| B) = 0.6 and P(B | A) = 0.75. a. P(A ∩ B)
b. P(B) c. P(A ∪ B) d. P(A ∩ B) e. P(B | A)

Let A and B be two independent events in the sample space S.
Which of the following statements
is/are true? Circle all that apply. [3 marks]
(a) The events A and Bc are independent.
(b) The events Ac and Bc are independent.
(c) The events (A \ B) and (Ac \ Bc) are independent.
(d) None of the above.

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