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Sum crazy dice (Posted on 2006-11-30) Difficulty: 3 of 5
a) I have a pair of fair n-sided dice. The probability when both are rolled that their results differ by two is the same as that the sum will be 5 or less. Find n.

b) I have two dice, one with n sides and the other with m sides. When they are rolled the probability they are equal is the same as that they sum to 13 or higher. Find n and m.

c) I have a trio of n-sided dice. When I roll them all the probability that the dice all show different numbers is greater than when they sum 15 or less but less than when they sum 16 or less. Find n.

Note: "x sided dice" are numbered with consecutive integers from 1 to x.

No Solution Yet Submitted by Jer    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): Part c) clarified. intended solution?? Comment 8 of 8 |
(In reply to re: Part c) clarified. by Charlie)

Part c works if "15 or less" is replaced by "less than 15", and similarly for 16, with the answer being n=8:

3             0.2222222222 1.0000000000 1.0000000000
4             0.3750000000 1.0000000000 1.0000000000
5             0.4800000000 0.9920000000 1.0000000000
6             0.5555555556 0.9074074074 0.9537037037
7             0.6122448980 0.7551020408 0.8367346939
8             0.6562500000 0.5937500000 0.6835937500
9             0.6913580247 0.4581618656 0.5418381344
10            0.7200000000 0.3520000000 0.4250000000
11            0.7438016529 0.2712246431 0.3328324568
12            0.7638888889 0.2106481481 0.2615740741

FOR n = 3 TO 12
  succDiff = 0: t15Ct = 0: t15suc = 0: t16Ct = 0: t16suc = 0
  FOR a = 1 TO n
  FOR b = 1 TO n
  FOR c = 1 TO n
    IF a <> b AND a <> c AND b <> c THEN
     success = 1
    ELSE
     success = 0
    END IF
    succDiff = succDiff + success
    t = a + b + c
    IF t < 15 THEN
      t15Ct = t15Ct + 1
      t15suc = t15suc + success
    END IF
    IF t < 16 THEN
      t16Ct = t16Ct + 1
      t16suc = t16suc + success
    END IF
  NEXT
  NEXT
  NEXT
  ways = n * n * n
  PRINT n,
  PRINT USING "##.##########"; succDiff / ways; t15Ct / ways; t16Ct / ways
NEXT

Edited on December 1, 2006, 3:04 pm
  Posted by Charlie on 2006-12-01 15:03:13

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