My aim is to become sharp in the necessary knowledge of basic probability and counting to follow my studies of Statistics for Computer Science.
Right now I found the following book:
http://aops-cdn.artofproblemsolving.com/products/intro-counting/toc.pdf
But the price is too high.
Are there any other cheaper alternatives that cover the same ground?
May be, this online course will be useful for you. It's free and high-quality one, one of the best on Coursera. It's running right now in its seventh week, but you still can join, watch lectures and do homeworks. Sorry to say, early homeworks won't be graded.
UPDATE from 8/16/2016:
Sorry to say, the free "Probability" course by Santosh Venkatesh (hyperlinked above) has been taken down by Coursera. Why? I have no idea. However, his book is available. Yes, it's expensive...