Question

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.

Answer #1

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

Consider two events A and B from the same sample space S.
Which of the following statements is not true:
a) If A and B are independent, then P left parenthesis A right
parenthesis equals P left parenthesis right enclose A B right
parenthesis
b) If A and B are mutually exclusive events, then P left
parenthesis A union B right parenthesis equals P left parenthesis A
right parenthesis plus P left parenthesis B right parenthesis
c) If A and...

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, B and C be events in a sample space, S. Suppose that S =
A ∪ B ∪ C, and that the following hold: A ∩ C = ∅, B ∩ C = ∅. Let
P(A) = 0.3, P(B) = 0.5 and P(C) = 0.5. Find P(A ∩ B).

Let S be a sample space with probability P and let A ⊂ S, B ⊂ S
be independent events. Given P (B) = 0.3 and P (A ∪ B) = 0.65, find
P (A).

Let A and B be two events from the sample space S. Given
P(A)=0.3, P(B)=0.4 and P(A or B)=0.6
Find: a) P(A and B) b) P(not A) c) P(not B) d) P(not(A and B))
e) P(not(A or B))

Let A1, A2, . . . , An be n independent events in a sample space
Ω, with respective probability pi = P (Ai). Give a simple
expression for the probability P(A1 ∪A2 ∪...∪An) in terms of p1,
p2, ..., pn. Let us now apply your result in a practical setting: a
robot undergoes n independent tests, which are such that for each
test the probability of failure is p. What is the probability that
the robot fails at least...

Suppose that A and B are two events in a sample space
S with P(B)=0.04, P(A n B)=0.03 and P(A u B) =0.46
a) what is P(B) complement
b)what is P(A)
c)what is P( AUB) complement
d)what is P(A/B)
e)what is P(A complement n B)
f)Are two events A and B independent? justify your answer.
g) Are two events A and B mutually exclusive? Justify your
answer.

Consider two events, A and B, of a sample space such that P(A) =
P(B) = 0.7 a).Is it possible that the events A and B are mutually
exclusive? Explain. b).If the events A and B are independent, find
the probability that the two events occur together. c).If A and B
are independent, find the probability that at least one of the two
events will occur. d).Suppose P(B|A) = 0.5, in this case are A and
B independent or dependent?...

1) Let S = {H, T} be the sample space associated to the fair
coin-flipping. Is {H} independent from {T}?
2) Let S = {HH, HT, TH, T T} be the sample space associated to
flipping fair coin twice. Consider two events A = {HH, HT} and B =
{HT, T H}. Are they independent?
3) Suppose now we have a biased coin that will give us head with
probability 2/3 and tail with probability 1/3. Let S = {HH,...

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