Well, since you've only posted the question and not shown any work of your own, I'll assume you're stuck the beginning and need a hint to even get started. Here's some questions to think about to hopefully get you going in the right direction:
1) You pick 6 numbers out of a possible 49. How many different combinations of numbers can you pick?
2) In order to win the largest prize the first six numbers must match your six numbers. What is the odds of that happening?
3) To get the answer to the previous question, think about the probability of the first number being a match. Then think about the probability of the second number being a match. Can you figure out the probability of both of those numbers being matches?
4) Let's say you had a much simpler system in which there were three prizes. If you have a 1/4 chance of winning each of those three prizes, then what is the probability of winning any one of the prizes? Can you apply this principle to your larger, more complex problem?
5) If you can figure out the probability of winning the largest prize, then what is the probability of not winning it? If you don't win the largest prize, then a seventh number must be drawn. What is the probability of winning that prize? And so on until you reach the 35th ball being drawn...
Also, if nothing else, you might want to check the wording of the problem - it's a bit confusing to me. At one point it says: "...and the smallest prize being awarded if you don't match your 6th and final number" but then later on it also says: "No prizes are awarded unless all 6 of your chosen numbers are matched." But these two statements contradict each other. If you don't win a prize unless all six of your numbers are matched, then how can you possibly win the smallest prize by not matching your sixth number?
