Overall Odds of Winning

If the overall odds of winning are 1 in 13 and I play once a week will I win 4 times a year?

In short, no.

Why not?

To make the maths simpler lets look at the chances of rolling a six when rolling a single dice 6 times:
On the first roll the chance of rolling a six is 1 in 6
So on the first roll there is a 56 chance we didn't roll a six and in that case we need to roll again.
On our second roll the chance of rolling a six is still 1 in 6, but our overall probability for the two rolls is:

16 + ( 56 x 16 ) = 1136 which is slightly less than 2 in 6.

Why did we multiply the odds for the second roll by 56? We did this because we only need to consider the second roll if we didn't roll a six on the first roll, so only in the 5 of 6 occassions that we didn't roll a six.
On the third roll there is a 2536 chance that we didn't roll as six on the previous two rolls, therefore the chances are:

16 + ( 56 x 16 ) + ( 2536 x 16 ) = 91216 which, again, is slightly less than 3 in 6

Another way at looking at this is each time we roll, there is a 56 chance it is not a six, so we can work out the chances of not rolling a six more easily:

56 x 56 x 56 x 56 x 56 x 56 which is just over 13

The general formula for rolling a six in n rolls is 1 - 56^n which means the chances of rolling a six in 6 rolls is only about 23, in fact although the probability rises with each roll it will never be 11 (i.e. you are never guaranteed to roll a six).

It works the same way for lottery numbers. Although the probability of a particular number appearing increases with each one drawn, it can never be guaranteed that it will appear within a certain number of draws.