Solution to simple Probability problem The problem:

Jan and 5 other children are in a classroom. The principal of the school walks in and chooses two children at random. What is the probability that Jan is chosen?

My not-so-straightforward solution:
I first solve for the probability that Jan is not chosen (for caning, I think).

Probability that Jan is not the first choice : 5/6
Probability that Jan is not the second choice : 4/5 (since there is one less person to choose from)
Probability that Jan is not the first choice AND not the second choice : 5/6 * 4/5 = 2/3
Therefore, Probability that Jan is either the first choice OR the second choice : 1 - (2/3) = 1/3

This answer is correct according to the book but, is the solution right ?

Your solution is right.
Probability Jan is chosen is $\frac{\binom{5}{1}}{\binom{6}{2}}=\frac{1}{3}$
