The probability that a student owns a dog is 1/3. the probability that the same student owns
a dog and a cat is 2/15 . determine the probability that the student owns a cat.

Let us call p the probability of occurence of the event D: "a student owns a dog"
and q the probability of occurence of the event C: "a student owns a cat"
In this exercice we have to assume that the events D and C are independent which means that none of them influence the occurence of the other. Because if they are not independent there really is noway to deduce q here. 
In reality those events would probably not be independent since if I had a dog, I would really think twice before buying a cat ;)

Well if they are independent then you know that  The Probability of occurence of D and C at the same time equals P(D)*P(C)=p*q
So we can deduce that 2/15=1/3*q
q=3*2/15
q=6/15
q=2/5

Once again this is theoritical