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A standard die is rolled six times. What is the probability that the product of all six rolls is divisible by 3? Express your answer as a common fraction.

 Apr 17, 2022
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It  may be easier to figure out the number of  products that  are not divisible  by 3

 

For this to  happen we  cannot have  any  roll =  3  or  6

 

So.......  we can roll either a 1 or a 2 or a 4 or a 5   on  any roll   (since none of these are  divisible  by 3 )

 

The number of  possible  rolls    in this case   =   4^6    (we can choose any of the  4 numbers above on each  roll)

 

And the number of possible  rolls in all cases =   6^6

 

So....the probability that  any  product is  divisible by 3 is

 

1 -   4^6  / 6^6   =

 

1  -  (2/3)^6 =

 

1 - 64/729   =

 

665 / 729

 

cool cool cool

 Apr 17, 2022

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