You are given four dice that have numbers on their faces accordingto the followi

Question

You are given four dice that have numbers on their faces accordingto the following list:
d1: 1, 2, 3, 9, 10, 11
d2: 0, 1, 7, 8, 8, 9
d3: 5, 5, 6, 6, 7, 7
d4: 3, 4, 4, 5, 11, 12

For each pair of dice, compute the probability that the first diewill have a higher value showing
than the second will, and vice versa. You can easily show yourresults in a 4 by 4 matrix where
the row represents one die and the column represents another.Because we assume that you
will toss two different dice, the diagonal of the matrix should beleft blank because it represents
matching a die against itself.) These dice have an interestingproperty – can you determine it?

Explanation / Answer

Okay, I'm not going to do the whole matrix for you, but I'lldemonstrate how. Lets see the probability that d1>d2. There are 36different combinations of rolls possible (6 * 6). That is, d1is 1, and d2 can be any of 6 faces. then d1 is 2, d2 is anyof 6...so 36 possible. How many ways is d1greater.  Let's use ordered pairs, first number is rollof d1, second number is roll of d2: (d1, d2). d1 isgreater for the following ordered pairs: (1,0), (2,0), (2,1),(3,0), (3,1),(9,0),(9,1),(9,7),(9,8),(9,8),(10, all), (11,all). So that makes 22 ways out of 36 possible. Prob d1>d2= 22/36. Do this for d1 vs. d3, d1 vs d4 and place these vertically inthe first column of the matrix, with an X in the slot occupied byd1 vs. d1.  The first column begins: X 22/36 d1 >d3 d1>d4 Keep doing this - the next column is d2 >d1                                                            X                                                         d2>d3                                                          d2>d4 fill in all 4 X 4 slots of the matrix. After that, see if you can spot the property. It isn'tthat simple, but it can be found. I don't want to spoilit, but if you try and still can't get it, send me a message andI'll give it to you.

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