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.

(In reply to Part c) clarified. by Jer)

If the statement of the problem is:

the probability that the dice all show different numbers is greater than that they sum 15 or less but less than that they sum 16 or less.

 (I had interpreted "when" to mean "on the condition that")

the following table holds:

n         p(all different) p(15 or less)  p(16 or less)
3             0.2222222222 1.0000000000 1.0000000000
4             0.3750000000 1.0000000000 1.0000000000
5             0.4800000000 1.0000000000 1.0000000000
6             0.5555555556 0.9537037037 0.9814814815
7             0.6122448980 0.8367346939 0.8979591837
8             0.6562500000 0.6835937500 0.7656250000
9             0.6913580247 0.5418381344 0.6241426612
10            0.7200000000 0.4250000000 0.5000000000
11            0.7438016529 0.3328324568 0.3981968445
12            0.7638888889 0.2615740741 0.3171296296

And it appears that no line of the table satisfies the conditions.

As fractions the table appears as:

3               6/  27     27/  27     27/  27
4              24/  64     64/  64     64/  64
5              60/ 125    125/ 125    125/ 125
6             120/ 216    206/ 216    212/ 216
7             210/ 343    287/ 343    308/ 343
8             336/ 512    350/ 512    392/ 512
9             504/ 729    395/ 729    455/ 729
10            720/1000    425/1000    500/1000
11            990/1331    443/1331    530/1331
12           1320/1728    452/1728    548/1728

The program has some unneeded lines, left over from the old interpretation, but the calculation should be valid as the appropriate variables are used in the calculations:

 

DEFDBL A-Z
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

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

 

 

Posted by Charlie on 2006-12-01 14:58:41