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 Sum crazy dice (Posted on 2006-11-30)
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)

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 re: Part c) clarified. | Comment 7 of 8 |
(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 1.00000000004             0.3750000000 1.0000000000 1.00000000005             0.4800000000 1.0000000000 1.00000000006             0.5555555556 0.9537037037 0.98148148157             0.6122448980 0.8367346939 0.89795918378             0.6562500000 0.6835937500 0.76562500009             0.6913580247 0.5418381344 0.624142661210            0.7200000000 0.4250000000 0.500000000011            0.7438016529 0.3328324568 0.398196844512            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/  274              24/  64     64/  64     64/  645              60/ 125    125/ 125    125/ 1256             120/ 216    206/ 216    212/ 2167             210/ 343    287/ 343    308/ 3438             336/ 512    350/ 512    392/ 5129             504/ 729    395/ 729    455/ 72910            720/1000    425/1000    500/100011            990/1331    443/1331    530/133112           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

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