A coffee machine may be defective because it dispenses the wrong amount of coffe

Question

A coffee machine may be defective because it dispenses the wrong amount of coffee (C) and / or it dispenses the wrong amount of sugar (S). The probabilities of these defects are: P(C) = 0.05, P(S) = 0.04, P(C and S) = 0.01 (Remark that P(C and S) means P(C S).) What proportions of cups of coffee have: at least one defect no defects Let X be a discrete random variable with the following probability distribution function. What are the expected mean and variation of the random variable X? Use the contingency table below to find the following questions. An automated phone call routing system serves three offices, numbered 1, 2, and 3. Three phone calls come in, one intended for each of the three offices. Unfortunately, the phone system is malfunctioning | it assigns one call to each office, but in a random order. Assume all such assignments are equally likely. Define labels for the outcomes of this experiment. Using these labels for sample points, list the sample space. Define the events A ={ office 1 receives its correct call} B = {office 3 receives its correct call} C = {no office receives its correct call} List A, B, and C in terms of the sample space defined above. Which pairs of these events are disjoint? Are A and B independent events? Why or why not? (Show your work.)

Explanation / Answer

(1a) P(at least one defect) =P(c)+P(s)-P(c and s)

=0.05+0.04-0.01

=0.08

(b) P(no defect)= 1-0.05-0.04+0.01 = 0.92

(2) E(X)= expected mean= x*f(x)

=0*(1/10)+1*(2/5)+2*(1/2)

=1.4

E(X^2) =x^2*f(x)

=0*(1/10)+1*(2/5)+2^2*(1/2)

=2.4

Var(X) = E(X^2) - [E(X)]^2 = 2.4-1.4^2=0.44

(3a) P(A|B)= P(A and B)/P(B) =40/70 =0.5714286

(b)

P(A|B') = P(A and B')/P(B') =30/80 =0.375

(c)

P(A' |B') = P(A' and B')/P(B') =50/80 =0.625

(d) Event A and B are not independent because P(A and B) is not equal to P(A)*P(B)

(4a) The outcomes are X1, X2 and X3 where X1 is the call come in office 1, X2 is the call come in office 2 and X3 is the call come in office 3

The sample space is {X1,X2,X3}

(b)A and C events are disjoint and B and C events are disjoint

(c) A and B are independent events because event A is not effect to event B

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