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Figuring a wonky die (Posted on 2006-04-24) Difficulty: 3 of 5
I have an unfair six sided die with the following known characteristics:

1) The expected value is the same as for a fair die;
2) The probability of rolling either 1 or 2 or 5 is the same as the probability of rolling either 4 or 6;
3) The probability of rolling either a 2 or 3 is 1/2;
4) If the die is rolled twice in a row, the probability of getting 6 both times is the same as getting a 2 and then a 5;
5) The probability of rolling a 4 is ten times that of rolling a 1;
6) The probability of rolling a 6 is twice that of rolling a 5.

What is the probability of each side? This can and should be solved by hand.

See The Solution Submitted by Jer    
Rating: 3.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution (Full) | Comment 3 of 5 |
Side  Probability
-----------------
  1     0.025
  2     0.300
  3     0.200
  4     0.250
  5     0.075
  6     0.150
       ------
        1.000     

The expected value is 3.5 (same as a normal die).


I labeled the sides a-f (a is side 1, b is side 2, etc).  From the information given, you can create the following equations:

(1) a + b + c + d + e + f = 1

(2) a + b + e = d + f

(3) b + c = 0.5

(4) f^2 = b * e

(5) d = 10a

(6) f = 2e

You can then combine these equations to find the proper values.  From (6) we know that f = 2e.  From (4) and (6) we know that b = 4e.  Substituting into (2) we find that a + 5e = d + 2e, or a + 3e = d. 

Combining this with (3) and (6) and substituting in (1) we get a + 0.5 + d + 3e = 1, or 2d + 0.5 = 1, so d = 0.25.

From (5) we then know that a = 0.025.  Knowing that a + 3e = d, we can figure out that e = 0.075.  Then from (6) we know that f = 0.15.  Finally from (4) we figure out that b = 0.3, and from (3) that c = 0.2.  Voila! 

The fact that the die has the same expected value as a fair die seemed to be of no use other than to confirm that the solution actually works.


Note: Sorry about the multiple posts, I'm not sure what happened. :)


  Posted by tomarken on 2006-04-24 13:36:49
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