Feller, "An Introduction to Probability Theory and Its Applications", Third Edition, gives a useful approximation on p. 325, equation 7.11.

Suppose we toss a possibly biased coin $n$ times, where the probability of a head is $p$ and $q = 1-p$.  Let $q_n$ be the probability there is no run of $r$ successive heads.  Then

$$q_n \sim \frac{1-px}{(r+1-rx)q} \cdot \frac{1}{x^{n+1}}  $$

where $x$ is the smallest positive root of $1 - x + q p^r x^{r+1} = 0$.

For your problem, we have $r = 5$, $p = 1/38$, and $q = 37/38$, from which we calculate $x \approx 1 + 1.228854 \times 10^{-8}$.  

It works out that $q_n = 1/2$ for $n \approx 5.64 \times 10^7$, i.e. it takes about 56 million trials to have a 50% chance of guessing correctly 5 times in a row.

If you attempt to predict a Roulette wheel $n$ times, what's the probability you'll get $5$ in a row at some point?
I've been using the same set of photographic portraits in my mockups for a long time, and I'm looking for a more extensive set of square, male and female avatars/photos (portraits).

What are good sets to use?
