Two fair die are tossed, and the uppermost face of each die is observed. The fol

Question

Two fair die are tossed, and the uppermost face of each die is observed. The following events are defined from this random experiment:

A represent the event the uppermost faces sum to five

B represent the event that the product of the uppermost faces is four. For example, die1*die2 = 4

C represent the event that the absolute difference between the uppermost faces is 1. For example, (die1die2)=1

Part (a) Find the probability that the uppermost faces do not sum to five

Part (b) Find P(AUC)

Part (c) What is the probability that the uppermost faces do not sum to five or are not a product of 4?

Part (d) Find P(A(BUC))

Part (e) Are the events a sum of 5 and a product of 4 mutually exclusive events? Select the most appropriate reason below.
A. A sum of 5 and a product of 4 are not mutually exclusive events because P(AB)0
B. A sum of 5 and a product of 4 are mutually exclusive events because they are not independent events.
C. A sum of 5 and a product of 4 are not mutually exclusive events because P(AB)P(A)P(B)
D. A sum of 5 and a product of 4 are mutually exclusive events because P(AB)=0
E. A sum of 5 and a product of 4 are not mutually exclusive events because P(AB)=P(A)P(B)

Explanation / Answer

here there are 4 events in event A ={(1,4),(4,1),(2,3),(3,2)}

a) probability that the uppermost faces do not sum to five =1-P( upper most faces have sum 5) =1-4/36=32/36 =8/9

b) P(AUC) =P( uppermost sum is 5 or absolte differnce is 1) =P(C) (as A is a subset of C)

=10/36=5/18

c) probability that the uppermost faces do not sum to five or are not a product of 4 =1-P( both a sum to five or product of 4) =1-2/36 =34/36 =17/18 (as there are two events (1,4) and (4,1) which satisfy both)

d)(P(An(BUC)) =as BUC will have all elements of A ; therefore P(An(BUC) =P(A) =4/36 =1/9

e)A. A sum of 5 and a product of 4 are not mutually exclusive events because P(AB)0

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