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