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#1 2007-07-31 12:54:23

mpthomas
Member
Registered: 2007-07-27
Posts: 7

Dice roll

If you throw a dice 6 times, what's the chance that you'd get a six on:

A: exactly one of the throws.
B: one or more of the throws.

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#2 2007-07-31 16:01:20

pi man
Member
Registered: 2006-07-06
Posts: 251

Re: Dice roll

How many different rolls of 6 dice are there?   6 dice with 6 different outcomes (1 through 6) each.  That would be 6^6 = 46656.

How many different ways to roll zero 6's?   6 dice with 5 different outcomes (1 through 5) or 5 ^ 6 = 15625.

How many different ways to roll one 6?   5 dice with 5 different outcomes (5^5 = 3125) plus one dice with one outcome (a six).  The six could be on any one of the six dice (6 choose 1).  So there are 3125 * 6 = 18750 ways to roll one dice.

Thats all you really need to know to compute the chances of rolling 1 six (18750 / 46656 =~ 40.2%) and the chance of rolling one or more sixes ((46656-15625) / 46656 =~ 66.5%) but let's continue anyway.

How many different ways to roll two 6's?   4 dice with 5 different outcomes (5^4 = 625).  And there are 15 different pairs of dice that the would roll the two 6's (6 choose 2).   625 * 15 = 9375. 

How many different ways to roll three 6's?   5^3 (3 dice with 5 outcomes each) * 20 (6 choose 3) = 2500

How many different ways to roll four 6's?  5^2 * 15 (6 choose 4)= 375

How many different ways to roll five 6's?  5^1 * 6 (6 choose 5) = 30

How many different ways to roll six 6's?  5^0 * 1 (6 choose 6) = 1

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#3 2007-07-31 16:06:05

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Dice roll

How many different ways to roll two 6's?   4 dice with 5 different outcomes (5^4 = 625).  And there are 15 different pairs of dice that the would roll the two 6's (6 choose 2).   625 * 15 = 9375.

How many different ways to roll three 6's?   5^3 (3 dice with 5 outcomes each) * 20 (6 choose 3) = 2500

How many different ways to roll four 6's?  5^2 * 15 (6 choose 4)= 375

How many different ways to roll five 6's?  5^1 * 6 (6 choose 5) = 30

How many different ways to roll six 6's?  5^0 * 1 (6 choose 6) = 1

All you need is how many ways to role no 6's and how many ways with no restrictions.  Then subtract.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2007-07-31 21:06:05

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Dice roll

Ricky wrote:

All you need is how many ways to role no 6's and how many ways with no restrictions.  Then subtract.

Pi Man already did that.

pi man wrote:

How many different rolls of 6 dice are there?   6 dice with 6 different outcomes (1 through 6) each.  That would be 6^6 = 46656.

How many different ways to roll zero 6's?   6 dice with 5 different outcomes (1 through 5) or 5 ^ 6 = 15625.

How many different ways to roll one 6?   5 dice with 5 different outcomes (5^5 = 3125) plus one dice with one outcome (a six).  The six could be on any one of the six dice (6 choose 1).  So there are 3125 * 6 = 18750 ways to roll one dice.

Thats all you really need to know to compute the chances of rolling 1 six (18750 / 46656 =~ 40.2%) and the chance of rolling one or more sixes ((46656-15625) / 46656 =~ 66.5%) but let's continue anyway.

How many different ways to roll two 6's?   4 dice with 5 different outcomes (5^4 = 625).  And there are 15 different pairs of dice that the would roll the two 6's (6 choose 2).   625 * 15 = 9375. 

How many different ways to roll three 6's?   5^3 (3 dice with 5 outcomes each) * 20 (6 choose 3) = 2500

How many different ways to roll four 6's?  5^2 * 15 (6 choose 4)= 375

How many different ways to roll five 6's?  5^1 * 6 (6 choose 5) = 30

How many different ways to roll six 6's?  5^0 * 1 (6 choose 6) = 1

neutral

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#5 2007-08-01 04:09:32

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Dice roll

Ah, missed it.  Thanks Jane.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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