THX A particular computer program outputs a number in {0, 2, 3} with probabiliti
ID: 3007324 • Letter: T
Question
THX
A particular computer program outputs a number in {0, 2, 3} with probabilities as follows: The program is run twice and its first and second output are recorded. Let A be the event that the first output is 0. Let D be the event that the second output is 2. Let C be the event that the sum of the two outputs is exactly 5. Find Pr(A), Pr(B) and Pr(C). Find Pr(A Intersection B), Pr(A Intersection C) and Pr(B Intersection C). Are A and C independent events? Are B and C independent events? Find Pr(A Union C) and Pr(B Union C).Explanation / Answer
(a)
Pr(A)=1/3
Pr(B)=1/2
C can occur if 2 comes on first output and 3 on second and vica versa
Hence, Pr(C)=1/2*1/6+1/6*1/2=1/6
(b)
Pr(A intersection B)=1/3*1/2=1/6
Pr(A intersection C)=0 because for C to occur 0 cannot be one of the outputs
Pr(B intersection C)=1/6*1/2=1/12
(c) For X and Y to be independent events we need
P(X intersection Y)=P(X)P(Y)
Hence A and C are not independent
P(B)P(C)=1/2*1/6=1/12
P(B intersection C)=1/12
Hence , B and C are independent events
(d)
Pr(A union C)=Pr(A)+Pr(C)-Pr(A intersection C)=1/3+1/6-0=1
Pr(B union C)=Pr(B)+Pr(C)-Pr(B intersection C)=1/2+1/6-1/12=7/12
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