If I understand you correctly, there will be 510 people in the contest once your friends have entered, and you want to know the likelihood that at least one of the 11 of you wins a book?

Think about it this way. There are 510 numbers in a hat, and 50 are drawn consecutively. What is the probability that none of the numbers 1, ..., 11 are drawn? For the first draw, this is 499/510. For the second, there is now one less number in the hat, but all of the numbers 1, ..., 11are still there. So the probability is 498/509. For the third draw, it is 497/508, and so on.

In all the probability that none of you gets a book is

(499.498...451.450)/(510.509....462.461) = approx. 0.3176.

So the likelihood that at least one of you wins a book is about 68.24%.