Page 76 - Probability and Statistical Inference

1. Notions of Probability 53

1.4.1 Prove the Theorem 1.4.1.
1.4.2 In the Theorem 1.4.2, prove that (ii) ⇒ (iii) ⇒ (iv).
1.4.3 Consider a sample space S and suppose that A, B are two events
which are mutually exclusive, that is, A ∩ B = ϕ. If P(A) > 0, P(B) > 0, then
can these two events A, B be independent? Explain.
1.4.4 Suppose that A and B are two events such that P(A) = .7999 and
P(B) = .2002. Are A and B mutually exclusive events? Explain.
1.4.5 A group of five seniors and eight first year graduate students are
available to fill the vacancies of local news reporters at the radio station in a
college campus. If four students are to be randomly selected from this pool
for interviews, find the probability that at least two first year graduate stu-
dents are among the chosen group.
1.4.6 Four cards are drawn at random from a standard playing pack of
fifty two cards. Find the probability that the random draw will yield
(i) an ace and three kings;
(ii) the ace, king, queen and jack of clubs;
(iii) the ace, king, queen and jack from the same suit;
(iv) the four queens.
1.4.7 In the context of the Example 1.1.2, let us define the three events:
E : The sum total of the scores from the two up faces exceeds 8
F : The score on the red die is twice that on the yellow die
G : The red die scores one point more than the yellow die
Are the events E, F independent? Are the events E, G independent?
1.4.8 (Example 1.4.7 Continued) In the Example 1.4.7, the prevalence of
the disease was 40% in the population whereas the diagnostic blood test had
the 10% false negative rate and 20% false positive rate. Instead assume that the
prevalence of the disease was 100p% in the population whereas the diagnostic
blood test had the 100α% false negative rate and 100β% false positive rate, 0 <
p, α, β < 1. Now, from this population an individual is selected at random and
his blood is tested. The health professional is informed that the test indicated
the presence of the particular disease. Find the expression of the probability,
involving p, α and β, that this individual does indeed have the disease. {Hint:
Try and use the Beyes’s Theorem.}
1.4.9 Let us generalize the scenario considered in the Examples 1.4.5-
1.4.6. The urn #1 contains eight green and twelve blue marbles whereas
the urn #2 has ten green and eight blue marbles. Suppose that we also
have the urn #3 which has just five green marbles. These marbles have
the same size and weight. Now, the experimenter first randomly selects
an urn, with equal probability, and from the selected urn draws one marble