# Six-Sided Dice Probability - Getting a 5 or 6

• Jan 18th 2013, 12:01 PM
Six-Sided Dice Probability - Getting a 5 or 6
What are the chances of four of X (variable) rolled six-sided dice producing a result of “5” or “6” in one roll?
4 dice = 1.2%
5 dice = ???
6 dice = ???
etc.

I know that each die has a 1 out of 3 (or 1/3, or 33.4%) chance of producing the desired result of 5 or 6 (also called a "hit"). I know that the chances of all four dice of a four-die roll "hitting" is equal to the chance of one of them hitting times the chance of each other one hitting, i.e.
(1/3)*(1/3)*(1/3)*(1/3)

• Jan 18th 2013, 12:17 PM
Plato
Re: Six-Sided Dice Probability - Getting a 5 or 6
Quote:

What are the chances of four of X (variable) rolled six-sided dice producing a result of “5” or “6” in one roll?
4 dice = 1.2%
5 dice = ???
6 dice = ???
etc.!

I simply do not understand.

Are you asking: If we roll seven dice, what is the probably getting at least one 5 or 6?

OR, Are you asking: If we roll seven dice, what is the probably getting at exactly four 5's or 6's?

Or is it some else all together?
• Jan 18th 2013, 01:20 PM
Re: Six-Sided Dice Probability - Getting a 5 or 6
If we roll seven dice, what is the probability of getting exactly four 5's or 6's.

Sorry for the ambiguity.
• Jan 18th 2013, 01:29 PM
Plato
Re: Six-Sided Dice Probability - Getting a 5 or 6
Quote:

If $\displaystyle N\ge 4$ is the number are dice rolled the probability of getting exactly four 5's or 6's is:
$\displaystyle \binom{N}{4}\left(\frac{2}{3}\right)^4\left(\frac{ 1}{3}\right)^{N-4}$
where $\displaystyle \binom{N}{4}=\frac{N!}{4!(N-4)!}$