If n dice are thrown then find the probability of the event A that the product of the numbers shown by thrown n dice is divisible by 4.

Question

I've found the following:
n(S) = 6^n
The product will be divisible if at least two of the numbers are a combination of the following:
(1 or 2 or 3 or 4 or 5 or 6,4)
or
(2,2)
or
(2,6)
I don't know how to put this information together

Answer 1

There are precisely two disjoint ways in which the product can fail to be a multiple of 4:

A) It could be that every number rolled is odd (i.e., is either 1 or 3 or 5).

B) It could be that all but one number rolled is odd, while the remaining value is even but not divisible by 4 (i.e., is either 2 or 6).

The probability of A is $(3/6)^n = 1/2^n$. (This is because there are $3$ ways to come out odd out of $6$ total equiprobable possibilities, on each of $n$ rolls).

The probability of B is $2/6 \times n \times (3/6)^{n - 1} = 1/3 \times n/2^{n-1} = 2/3 \times n/2^n$. (The factor of $2/6$ from the two choices out of six for the even value, the factor of $n$ for the choice of which particular roll out of $n$ total takes on the even value, and the factor of $(3/6)^{n - 1}$ for the probability of the remaining $n - 1$ rolls coming out odd, as above)

Combining these, the probability we are interested in (the probability that neither of these two disjoint events happens) is $1 - (1 + 2n/3)/2^n$.

Answer 2

I made a spreadsheet-one of the first things you should do for problems like this. The column $n$ is the number of dice and the next four columns are the number of combinations with the sum equal to the header $\bmod 4$. The total column is a check. It sums the row and should equal $6^n$. I find $$\begin {array} {r r r r r}n&0&1&2&3&total\\1&1&2&2&1&6\\2&9&8&9&10&36\\3&55&55&53&53&216\\4&322&324&326&324&1296\\5&1946&1942&1942&1946&7776\\6&11664&11668&11664&11660&46656\\7&69980&69980&69988&69988&279936\\8&419912&419904&419896&419904&1679616\\9&2519416&2519432&2519432&2519416&10077696\\10&15116544&15116528&15116544&15116560&60466176\\11&90699280&90699280&90699248&90699248&362797056\\12&544195552&544195584&544195616&544195584&2176782336\\13&3265173536&3265173472&3265173472&3265173536&13060694016\\14&19591041024&19591041088&19591041024&19591040960&78364164096 \end {array}$$ The curious thing is that the differences between the columns are always a power of $2$, the pattern of the differences repeats with period $8$ and grows by a factor $2^4$ in $8$ rows. You should be able to justify all those observations by induction. Think of a column vector of the numbers in my table, a matrix multiply to go from one row to the next, and the eigenvalues/eigenvectors of that matrix.