What is the probability of rolling a pair of six sided dice five times and getting four 4's?

The
probability of rolling a pair of six sided dice five times and getting four 4's is exactly the
same as rolling one six sided die ten times and getting four 4's.

Each die
has the same probability of being a 4, ie 1/6 and the same probability of not being a 4, ie 5/6.
To get a combination of four 4's and 6` ` 'not 4's ' the probability is (1/6)^4 x (5/6)^6 as we
multiply the individual probabilities together.

The four 4's could occur on
any of the 10 dice rolls, and there are what we term '10 choose 4' ways of choosing these four
4's out of the 10 dice rolls:

10 choose 4 = `(10!)/(4!6!) =
(10.9.8.7.6.5.4.3.2.1)/((4.3.2.1)(6.5.4.3.2.1)) = (10.9.8.7)/(4.3.2.1) = (10.3.7)/1 = 210`

So there are 210 configurations that we can have of our four 4's and 6 'not
4's '.

We can rewrite this as `(10/1) (9/2)(8/3) (7/4)` which is 'ten ways of
choosing the first 4, nine ways of choosing the second, eight ways of choosing the third and
seven ways of choosing the fourth. Once we choose a 4, we can only choose the other 4's out of
the remaining dice rolls.

The probability of four 4's from five
rolls of two six sided dice is

210 x (1/6)^4 x
(5/6)^6 = 0.054