SOLUTION: Nonstandard dice can produce intercst- ing distributions of outcomes. Suppose you have two bal- anced, six-sided dice. Die A has faces with 2, 2, 2, 2, 6, and 6 spots. Die B has t

Question 678873: Nonstandard dice can produce intercst- ing distributions of outcomes. Suppose you have two bal- anced, six-sided dice. Die A has faces with 2, 2, 2, 2, 6, and 6
spots. Die B has three faces with 5 spots and three faces with 1spot. Imagine that you roll both dice at the same time.
(a). Find a probability model for the difference (Die A - Die B) in the total number of spots on the up-faces.
(b). Which die is more likely to roll a higher number? Justify your answer.
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!
(a). Find a probability model for the difference (Die A - Die B) in the total number of spots on the up-faces.

Here are all 36 possible rolls (A,B) with these weird dice:

(2,5)  (2,5)   (2,5)  (2,1)   (2,1)  (2,1)
 
(2,5)  (2,5)   (2,5)  (2,1)   (2,1)  (2,1)
 
(2,5)  (2,5)   (2,5)  (2,1)   (2,1)  (2,1)
 
(2,5)  (2,5)   (2,5)  (2,1)   (2,1)  (2,1)
 
(6,5)  (6,5)   (6,5)  (6,1)   (6,1)  (6,1)
 
(6,5)  (6,5)   (6,5)  (6,1)   (6,1)  (6,1)

Here are all the differences (Die A - Die B)
in the corresponding positions to the above:

 -3     -3      -3      1      1      1

 -3     -3      -3      1      1      1

 -3     -3      -3      1      1      1

 -3     -3      -3      1      1      1

  1      1       1      5      5      5

  1      1       1      5      5      5

There are 12 -3's, 18 1's, and 6 5's.  And there
are 36 possible rolls, so each probability is the
number of ways to roll the difference over 36.

So we list the probablity distribution function:

(Die A - Die B)     Prob. of sum     
        x               P(x)           
-------------------------------------
       -3            12/36 = 1/3           
        1            18/36 = 1/2           
        5             6/36 = 1/6           
-------------------------------------

(b). Which die is more likely to roll a higher number? Justify your answer.

Relisting all 36 possible rolls (A,B):
 
(2,5)  (2,5)   (2,5)  (2,1)   (2,1)  (2,1)
 
(2,5)  (2,5)   (2,5)  (2,1)   (2,1)  (2,1)
 
(2,5)  (2,5)   (2,5)  (2,1)   (2,1)  (2,1)
 
(2,5)  (2,5)   (2,5)  (2,1)   (2,1)  (2,1)
 
(6,5)  (6,5)   (6,5)  (6,1)   (6,1)  (6,1)
 
(6,5)  (6,5)   (6,5)  (6,1)   (6,1)  (6,1)

We list which one shows the higher number, die A or die B,
in the corresponding positions to the above rolls:

  B      B       B      A      A      A

  B      B       B      A      A      A

  B      B       B      A      A      A

  B      B       B      A      A      A

  A      A       A      A      A      A

  A      A       A      A      A      A

die A has the higher number in 24 rolls and die  
B has the higher number in only 12 rolls.  So the
probability that A has the higher number is 24/36
or 2/3 and the probability that A has the higher
number is only 12/36 or 1/3.

Edwin

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