Question

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 one of the tests?

Answer #1

P( A_{1} U A_{2} U....... U A_{n} ) =
p_{1} + p_{2} + p_{3} + ..... +
p_{n}

and

Probability that the robot fails atleast one of the tests =( 1 -
( 1 - p )^{n} )

consider a sample space defined by events a1, a2, b1 and b2
where a1 and a2 are complements .given p(a1)=0.2 p(b1/a1) = 0.5 and
p(b1/a2) =0.7 what is the probability of p (a1/b1)
P(A1/B1)=
round to the 3rd decimal

Events A1, A2, and A3 form a partiton of sample space S with
Pr(A1)=27, Pr(A2)=47, and Pr(A3)=17. E is an event in S
with Pr(E|A1)=35, Pr(E|A2)=25, and Pr(E|A3)=15.
What is Pr(E)?
What is Pr(A1|E)?
What is Pr(E′)?
What is Pr(A′1|E′)?
Enter your answers as whole numbers or fractions in lowest
terms.

Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q

Let S = {(a1, a2, . . . , an)| n ≥ 1, ai ∈ Z ≥0 for i = 1, 2, .
. . , n, an 6= 0}. So S is the set of all finite ordered n-tuples
of nonnegative integers where the last coordinate is not 0. Find a
bijection from S to Z +.

Events A1,A2, and A3 form a partition of sample space
S with Pr(A1)=3/7, Pr(A2)=3/7, Pr(A3)=1/7. E is an event in S with
Pr(E|A1)=3/5, Pr(E|A2)=2/5, and Pr(E|A3)=3/5.
What is Pr(E)?
What is Pr(A2|E)?
What is Pr(E')?
What is Pr(A2'|E')?

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.

Let S = {(a1,a2,...,an)|n ≥ 1,ai ∈ Z≥0 for i = 1,2,...,n,an ̸=
0}. So S is the set of all finite ordered n-tuples of nonnegative
integers where the last coordinate is not 0. Find a bijection from
S to Z+.

Let A1, ... ,20 be independent events each with probability 1/2.
Let X be the number of events among the first 10 which occur and
let Y be the number of events among the last 10 which occur. Find
the conditional probability that X = 5, given that X + Y = 12.

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

Q.3. (a) Let an experiment consist of tossing two standard
dice. Define the events, A = {doubles appear} (That is (1, 1), (2,
2) etc..)
B = {the sum is bigger than or equal to 7 but less than or
equal to 10}
C = {the sum is 2, 7 or 8}
(i) Find P (A), P (B), P (C) and P (A ∩ B ∩ C). (ii) Are
events A, B and C independent?
(b) Let the sample space...

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