Question

Prove

(a). Events A and B are independent if and only if A^{c}
and B^{c} are independent.

(b). If events A and B both have a positive probability and are disjoint, then they cannot be independent.

Answer #1

If A and B are independent events , prove that :
1\ A and B are independent
2\ Ac and B are independent
4\Ac and Bc are independent

Give a mathematical derivation of the formula P((A ∩ Bc ) ∪ (Ac
∩ B)) = P(A) + P(B) − 2P(A ∩ B). Your derivation should be a
sequence of steps, with each step justified by appealing to one of
the probability axioms.
##### solution ##########
1 Since the events A ∩ Bc and Ac ∩ B are disjoint, we have,
using the additivity axiom, P((A ∩ Bc ) ∪ (Ac ∩ B)) = P(A ∩ Bc ) +
P(Ac...

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

Prove: P(A ∪ B ∪ C) = P(A) + P(Ac ∩ B) + P(Ac ∩ Bc ∩ C)

Prove that given △ABC and △A′B′C′, if we have AB ≡ A′B′ and
BC≡B′C′,then B<B′ if and only if AC<A′C′. You cannot use
measures.

Prove using only the axioms of probability that if A and B are
events and A ⊂ B, then P(Ac ∩ B) = P(B) − P(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.

Prove that if A*B*C, then ray AB = ray AC and ray BC is a subset
of ray AC

1. (a) Let a, b and c be positive integers. Prove that gcd(ac,
bc) = c x gcd(a, b). (Note that c gcd(a, b) means c times the
greatest common division of a and b)
(b) What is the greatest common divisor of a − 1 and a + 1?
(There are two different cases you need to consider.)

Two independent events A and B are given, and P(Bc|A ∪ B) = 1/3,
P (A|B) = 1/2. What is P (Bc)?

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