2) A triangular die is a four-sided die, each side possessing either a number 1,

Question

2) A triangular die is a four-sided die, each side possessing either a number 1, 2, 3, or 4. Two such die are tossed simultaneously and the bottom faces - the face that each die lands on - is observed. Consider the following events: AA the bottom-most faces sum to six BB each bottom-most face shows an even number CC both bottom-most faces show the same number, referred to as doublesdoubles

Part (a) What is the probability that the bottom-most faces do not sum to six? (Use four decimals in your answer) Part (b) Find P(AB)P(AB) (Use four decimals)

Part (c) Find P(ACCc)P(ACCc) (use four decimals)

Part (d) Find P(ABC)P(ABC) (use four decimals)

Part (e) Are the events AA and CC mutually exclusive events? Select the most appropriate reason below.

A. AA and CC are mutually exclusive events because they are not independent events.

B. AA and CC are mutually exclusive events because P(AC)=0P(AC)=0.

C. AA and CC are not mutually exclusive events because P(AC)=P(A)P(C)P(AC)=P(A)P(C).

D. AA and CC are not mutually exclusive events because P(AC)P(A)P(C)P(AC)P(A)P(C).

E. AA and CC are not mutually exclusive events because P(AC)0P(AC)0

Explanation / Answer

(a) There are 4*4 = 16 different options for the bottom faces.

Of these 2+4, 4+2, 3+3 are the only ones that sum to 6.

A = {(2,4),(4,2),(3,3)}

Remaining 13 options do not sum to 6.

=> Probability that the bottom faces do not sum to 6 P(Ac) = 13/16 = 0.8125.

(b) A U B is the event that the bottom faces sum of the bottom faces is 6 or the bottom faces are both even.

The only ways bottom faces are both even are (2,2) , (2,4) , (4,2) , (4,4).

P(B) = 4/16 = 0.25.

Thus A U B = {{2,2},{2,4},{3,3},{4,2},{4,4}}

P(A U B) = 5/16 = 0.3125.

(c) Bottom faces can show the same number in the cases (1,1),(2,2),(3,3),(4,4). Thus P(C) = 4/16 = 0.25.

A U C = {(1,1),(2,2),(2,4),(3,3),(4,2),(4,4)}

P(A U C) = 6/16

P(Ac Cc) = P(A U C)c = 1 - 6/16 = 10/16 = 0.625.

(d) A B C = { }

P(A B C) = 0

(e) A C = {(3,3)}

=> P(A C) 0

Therefore they are not mutally exclusive.

The correct answer is E.

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