If a license plate has no numbers, what is the probability of it being a vanity plate? I have this theory that a license plate with no digits has a high probability of being a vanity plate. Lets assume license plates are $9$ characters, and are chosen by random unless they're a vanity plate. I figure that the probability of a license having no digits if chosen by chance is $\left(\frac{26}{36}\right)^9$, or about $5\%$. Is my math for that correct?
Given that probability, is there a way to determine the probability that a license plate with no digits was randomly chosen and is not a vanity plate?
 A: No. At the very least, you also need to know the a priori probability that a plate is a vanity plate. That is, when you ask:

is there a way to determine the probability that a license plate with no digits was randomly chosen and is not a vanity plate?

you are asking for the conditional probability $P(V|N)$ with $V$ being the event of the plate being a vanity plate, and $N$ being the event of the plate having no digits.  
Now, you could of course try to use Bayes' formula:
$$P(V|N)=\frac{P(N|V)\cdot P(V)}{P(N)}=\frac{1\cdot P(V)}{P(N)}= \frac{ P(V)}{P(N)}$$ 
But, like I said, you'd need to know the a priori probability $P(V)$ to use this.
Moreover, the $P(N)$ that this formula refers to is not the same as the $P(N)$ that you calculated: the $P(N)$ you calculated was the probability of getting no digits on a plate where all symbols are picked randomly, i.e. this was assuming no one has requested a vanity plate! 
Now, here is something you can do: find the actual $P(N)$, i.e. find the percentage of plates that are actually out there being used that have no numbers.  If you find that this is far more than that $5$% that you would get when creating plates randomly, then you have good reason to believe that lots of people got vanity plates.  If, however, you find that this $P(N)$ is close to that $5$%, then that suggest not many people order vanity plates. So .. start counting! :)
A: You are right about the probability of a randomly-generated 9-"letter" plate having no digits (if the "letters" are 0-9 and A-Z), but there's no way to know how an all-letter plate was chosen. I might have requested the vanity plate YHEOEBNWX. Maybe there are only 2 vanity plates in your state, or maybe there are millions. The fraction of all-letter plates that were human-selected is not something that can be determined mathematically.

Added: Also, there is the practical problem of how randomly-selected plates can avoid matching already-issued plates, so in practical terms, random generation of license plates is, well, not practical, at least not without some collision-avoiding rules, which may come into play more (or less) often for all-letter plates than for other plates.
A: Bayes' formula is the way to go. But you do first need to know how common vanity plates are in general. To illustrate that you need to know this, consider the two extremes: no one has vanity plates, at all, or everyone had vanity plates. In these cases, the probability of a digitless plate to be vanity is still either $0$ or $1$.
A: We can't do this without knowing probability of vanity plates and what proportions of vanity plates have no numbers.  
If $0.1\%$ of cars have vanity plates and half of them have no numbers then then $.999*(\frac {26}{36})^9\approx .05$ of the cars are no numbered standards and $.001*.5 = .0005$ are no numbered vanity then the probability of a no numbered plate being vanity is $\approx \frac {0.0005}{0.0505}\approx 1\%$.
But of $25\%$ of the cars have vanity plates and $90\%$  have no numbers then there are $.75*(\frac {26}{36})^9 \approx 4\%$ of the cars have standard issue plates with no numbers and $.25*.9=22.5\%$ of the cars have vanity plates with no numbers so the probality a plate with no numbers is $\approx \frac {22.5}{26.5}\approx 85\%$.
So that makes a huge difference how common vanity plates are.
Rule of thumb if a vanity plate with no numbers is more likely than five percent then it is likely the plate is vanity but if a vanity plate with no numbers is less likely than five percent it is likely or not.
A similare example is if you flip a coin five times and got all heads.  That a one in thirty-two probability.  That's unlikely, maybe you have a fake coin.  Well, fake coins are really rare... a lot fewer than one in thirty-two so you probably have a good coin and it was just a fluke.  But if you flip it $20$ times in a row and got all heads, the probability is less than one in a million.  Okay, we don't know how likely it is that you got a bad coin but it's probably higher than one in a million so this probably a fake coin.  
(On the other hand this event, flipping 20 heads and not knowing if the coin was fake, has probably never happened to you or anyone you know.)
If we don't know the likelihood of vanity plates in general, $5\%$ likelihood of a standard issue plate being all numbers just isn't rare enough for us to suspect a vanity plate unless vanity plates without numbers are more than $5\%$ of the cars as well.
