Answer:
If we take "at random" to mean uniformly at random (which is the typical interpretation when there is no other information given) then the answer is actually FIVE out of eleven (not 4 out of eleven as the other answers give).
In English, the letter Y is considered a vowel when it makes the sound of a long E as it does in the word PROBABILITY. So the vowels are the A, the two I's, the O, and the Y -- a total of five of the eleven letters.
The only thing I have to add is that while "at random" is usually taken to mean uniformly at random, it need not mean this. For example, we could assign the probabilities in such a way that each letter is twice as likely as the letter before it. So "p" would have probability 1/2047 of being selected, "r" would have probability 2/2047, "o" would have probability 4/2047, and so on. Finally "y" would have probability 1024/2047. In this case, we'd add the probabilities of the five vowels to get a total probability of 1346/2047.