Quickly calculate the probability of 12 dice having sum less than 30

I try to quickly answer the question of whether one should play a game: roll 12 fair dice and sum up the face values; if the sum is less than 30, win 10 dollars, otherwise lose 1 dollar.

Let $S_{12}=\sum\limits_{i=1}^{12} x_i$ be the sum where $x_i$ is the value for the ith die.

I think if the expected gain is greater than 0, we should play the game, and the gain is$$E = 10 \times P(S_{12}<30) - 1\times [1-P(S_{12}< 30)] >0 ?$$

So I tried to quickly calculate or approximate $P(S_{12}<30)$. My way is through CLT,$$\frac{S_{12} - 12E[x_i]}{\sqrt{12 var(x_i)} } \xrightarrow{d} N(0,1)$$$$\Rightarrow P(S_{12} < 30) \approx p(z < \frac{30-42}{\sqrt{35}}) = \Phi \left( \frac{30-42}{\sqrt{35}} \right) $$

But this approach still seems to require a certain amount of computation. Especially, it requires evaluating the normal cdf $\Phi(z)$.

So I was wondering if there is a better/quicker way without referring to computers/calculators.

Thanks in advance.