Math Is Fun Forum

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.

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

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..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Dice roll

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.

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

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..."