Let D1, D2, D3 be three four-sided dice whose sides have

Chapter 1, Problem 10E

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QUESTION:

Let \(D_{1}, D_{2}, D_{3}\) be three four-sided dice whose sides have been labeled as follows:

                 \(D_{1}: 0333 \quad D_{2}: 2225 \quad D_{3}: 1146\)

The three dice are rolled at random. Let \(A, B\), and \(C\) be the events that the outcome on die \(D_{1}\) is larger than the outcome on \(D_{2}\), the outcome on \(D_{2}\) is larger than the outcome on \(D_{3}\), and the outcome on \(D_{3}\) is larger than the outcome on \(D_{1}\), respectively. Show that (a) \(P(A)=9 / 16\), (b) \(P(B)=9 / 16\), and (c) \(P(C)=10 / 16\). Do you find it interesting that each of the probabilities that \(D_{1}\) "beats" \(D_{2}, D_{2}\) "beats" \(D_{3}\0, and \(D_{3}\) "beats" \(D_{1}\) is greater than \(1 / 2\)? Thus, it is difficult to determine the "best" die.

Equation Transcription:

 

 

.

Text Transcription:

D_1:0333 D_2:2225 D_3:1146  

A,B C  

D_1  

D_2

D_3

P(A)=9/16

P(B)=9/16

P(C)=10/16.  

1/2

Questions & Answers

QUESTION:

Let \(D_{1}, D_{2}, D_{3}\) be three four-sided dice whose sides have been labeled as follows:

                 \(D_{1}: 0333 \quad D_{2}: 2225 \quad D_{3}: 1146\)

The three dice are rolled at random. Let \(A, B\), and \(C\) be the events that the outcome on die \(D_{1}\) is larger than the outcome on \(D_{2}\), the outcome on \(D_{2}\) is larger than the outcome on \(D_{3}\), and the outcome on \(D_{3}\) is larger than the outcome on \(D_{1}\), respectively. Show that (a) \(P(A)=9 / 16\), (b) \(P(B)=9 / 16\), and (c) \(P(C)=10 / 16\). Do you find it interesting that each of the probabilities that \(D_{1}\) "beats" \(D_{2}, D_{2}\) "beats" \(D_{3}\0, and \(D_{3}\) "beats" \(D_{1}\) is greater than \(1 / 2\)? Thus, it is difficult to determine the "best" die.

Equation Transcription:

 

 

.

Text Transcription:

D_1:0333 D_2:2225 D_3:1146  

A,B C  

D_1  

D_2

D_3

P(A)=9/16

P(B)=9/16

P(C)=10/16.  

1/2

ANSWER:

Step 1 of 6:

Given that D1,D2 and D3 are the three four-sided dice. Sides of D1 are labeled  as 0,3,3,3. Sides of D2 are labeled as 2,2,2,5 and sides of D3 are labeled as 1,1,4,6.

The three events A,B,and C are defined as ,

A= outcome on D1 is larger than outcome on D2.

B=outcome on D2 is larger than outcome on D3.

C=outcome on D3 is larger than outcome on D1.


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