Probability and independence I am trying to solve a particular probability question. 
I have a fair 10-sides die, whose sides are labelled 1 through 10. I am trying to find the probability of rolling a multiple of 5 or an odd number. 
I find the probability as: 
P(multiple of 5) OR  P(odd number)=P(multiple of 5) + P(odd number)-[P(multiple of 5) AND P(odd number)]=2/10+5/10-[(2/10)(5/10)]=6/10 (which is the correct answer)
Notice that I used the assumption that rolling a multiple of 5 is independent of rolling an odd number, since I essentially multiplied their probabilities to get the answer. However, rolling a multiple of 5 INVOLVES rolling an odd number in one case, namely rolling a 5. 
So did I get the correct answer by fluke? Can the multiplication rule also be used to describe non-independent events sometimes?

It might seem odd, but in fact the two events of rolling a multiple $5$ and rolling an odd number are independent.
To see this, let $A$ be the event that a multiple of $5$ is rolled, and $B$ be the event that an odd number is rolled.  What is $P(A)$?  What is $P(A|B)$?  Since the two numbers are the same, it follows that the outcome of $B$ does not affect that of $A$, so the events are independent.
As long as events $A$ and $B$ are independent, then $P(A\text{ or }B)=P(A)+P(B)-P(A)P(B)$, so you figured correctly.  The only part you were missing is proving that $A$ and $B$ are in fact independent.
Of course, the easiest way to solve this problem is to just list possibilities.  $6$ of the $10$ sides are odd numbers or multiples of $5$, so the probability is $6/10$.
Finally, I should mention that the multiplication rule only holds for independent events, because one way to define independence is by the formula $P(A\text{ and }B)=P(A)P(B).$
