Question

Two, 9-sided, fair die are tossed, and the uppermost face of each die is observed. The following events are defined from this random experiment: Let...

A : represent the event the uppermost faces sum to four

B : represent the event that the absolute difference between the uppermost faces is 2. For example, |die1-die2|=2

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

What is the probability that the top sides of these two dice:

Part (a) do not sum to four?

(If rounding, use at least four decimals in your answer)

Part (b) sum to four or the absolute difference of the top sides is equal to 2 ?

(If rounding, use at least four decimals in your answer)

Part (c) Find P(A^{c} ∪ B^{c})=

(If rounding, use at least four decimals in your answer)

Part (d) Are the events a sum of 4 and a product of 4 mutually exclusive events? Select the most appropriate reason below.

A. A sum of 4 and a product of 4 are mutually exclusive events because they are not independent events.

B. A sum of 4 and a product of 4 are not mutually exclusive events because P(A ∩ C) ≠ P(A) P(C).

C. A sum of 4 and a product of 4 are not mutually exclusive events because P(A ∩ C)=P(A) P(C).

D. A sum of 4 and a product of 4 are not mutually exclusive events because P(A ∩ B) ≠ 0.

E. A sum of 4 and a product of 4 are mutually exclusive events because P(A ∩ B)=0.

F. A sum of 4 and a product of 4 are not mutually exclusive events because P(A ∩ C) ≠ 0.

G. A sum of 4 and a product of 4 are not mutually exclusive events because P(A ∩ B) ≠ P(A) P(B).

H. A sum of 4 and a product of 4 are mutually exclusive events because P(A ∩ C)=0.

I. A sum of 4 and a product of 4 are mutually exclusive events because they are not independent events.

J. A sum of 4 and a product of 4 are not mutually exclusive events because P(A ∩ B)=P(A) P(B).

Part (e) Find P((A ∪ B) ∩ C)=

(If rounding, use at least four decimals in your answer)

Answer 1

The total number of outcomes (n) in this case are = 9² = 81. The events A, B and C can be written as:

A = { (1,3), (2,2), (3,1) }

B = { (1,3), (2,4), (3,1), (3,5), (4,2), (4,6), (5,3), (5,7), (6,4), (6,8), (7,5), (7,9), (8,6), (9,7) }

C = { (1,4), (2,2), (4,1) }

Let n(A), n(B) and n(C) denote the number of cases favourable to events A, B and C respectively. Thus, n(A) = 3, n(B) = 14 and n(C) = 3.

Part a) P(top sides of the two die do not sum to 4) = 1 - P(top sides sum to 4)

   = 1 - P(A) = 1 - 0.0370 = 0.9630

Part b) P(top sides of the two die sum to 4 or the absolute difference is equal to 2) = P(A or B) = P(A+B)

We know that,

Now,

Part c) We know that

  

Part d) Two events are said to be mutually exclusive if their intersection is . In this case, the events 'sum of 4' and 'product of 4' are the events A and C respectively.

Therefore, the reason F is the most appropriate answer i.e., sum of 4 and product of 4 are not mutually exclusive events because .

Part e)

Hence,