answer only question no. 3 What components do you need in order to build a proba

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answer only question no. 3

What components do you need in order to build a probabilistic model? Explain. Let A_1, ... A_n, be disjoint events that form a partition of the sample space, and assume that P(A_i) > 0 for all i = 1... n. Write down the general statement of probability of the event B (illustrated in the figure below for n = 3) If event B is definite evidence of a disease an A_1, .... A_3, are probable causes, write the expression for the probability that A_2 is the cause? An experiment involves two successive rolls of a fair four-sided dice. Let the event A = {the 1^st roll is 1} and event B = {the sum of two rolls is 4}. Are the events A and B independent? Next, you are given additional information that the event D = {the outcome of the two rolls were different} has occurred. Are events A and B conditionally independent of event D? A student crossed 30 plants of a species with Red flowers with 30 plants that have White flowers and obtained 40 offspring. Among the offspring only 10 were red. If a different student carried out a similar cross with 5 pairs of Red- and white-flowered plants and obtained 10 offspring, what the probability that none of the 10 offspring have Red flowers?

Explanation / Answer

Q.3 Total events = 4 * 4 = 16

Event A sample space = (1,1) ,(1,2) ,(1,3) ,(1,4)

Event B sample space = (1,3),(2,2) ,(3,1)

So P( A and B) = (1,3)= 1/16

P(A) . P(B) = (4/16) * (3/16) = 3/64

P(A and B) P(A) P(B) so not independent of each other

sample space of event D = (1,2), (2,1), (1,3), (3,1), (1,4), (4,1), (2,3), (3,2), (2,4), (4,2), (3,4), (4,3)

so P(D) = 12/16 = 3/4

The event D will be conditionally independent if

Pr (A B l D) = Pr(A l D) Pr(B l D)

Pr (A B l D) = 1/12

Pr(A l D) = 3/12 = 1/4

Pr(B l D) = 2/12 = 1/6

so Pr (A B l D) Pr(A l D) Pr(B l D)

so event D is not conditionally independent of A and B.

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