How to calculate the P(the sum of upward face after 7 dice rolls <= 30)

Question

I'm new to probability and am trying to learn the concept of calculating the sum of number after certain number of dice roll.

Could someone please give me a hint as to how to calculate the $$P(\text{the sum of upward face after }7\text{ dice rolls} \le 30)$$

Answer 1

If you have access to powerful tools you can enumerate all possible (and equiprobable) outcomes -- there are $6^7=279\,936$ of them. Then filter by total, there remains $253\,656$ outcomes, The probability is therefore $$ \frac{253656}{279936} = 0.906121. $$ Here is the Mathematica code to do that :

x = Tuples[{1, 2, 3, 4, 5, 6}, 7];
n = Length[x]
m = Length[Select[x, Total[#] <= 30 &]]
m/n // N

If not, the Central limit theorem will give you an approximation as follows. Define a random variable that equiprobably takes the values 1 to 6. Basic computations will tell you that $E(X)=3.5$ and $\sigma_X=1.70783$. The sum ($S$) over 7 trials approximately follows a normal distribution with mean $7\times 3.5$ and standard deviation $\sqrt{7}\times 1.70783$. You then need to compute (using tables or Excel for instance) $$ P(S\leq 30.5)=0.907891 $$ which is a pretty decent approximation.

Answer 2

Here is one way to reduce the computation substantially here. Assume that you have all $6's$ and subtract all cases with sum > 30, using stars and bars Th 1

Since dice sum pattern is symmetrical,it is the same as cases where sum $\le 18$

You will need to apply inclusion-exclusion on a minor scale to exclude a die face showing $\gt 6$

sum $7: \binom66 =1$
sum $8: \binom76 =7$
$...$
sum $12: \binom{11}6 = 462$
$...$
sum $13: \binom{12}6 - \binom71\binom66 = 917$
$...$
sum $18: \binom{17}6 - \binom71\binom{11}6= 9142$

overall sum = 26,280

$Pr = 1 - 26,280/6^7$