If you choose six numbers then the probability that one of them is the same as the first ball drawn is 6/49. Given that the first number is chosen correctly then the probability for drawing the second number correctly is 5/48. The probability of choosing all six numbers correctly is:

6 / 49 x 5 / 48 x 4 / 47 x 3 / 46 x 2 / 45 x 1 / 44 = 1 / 13,983,816

So, for example, if you have one attempt per week then you could expect to win, on average, once during the next 268,920 years!

Calculating The Odds explained from www.howstuffworks.com
Let's take a look at how to calculate the odds of picking the right number for a typical Lotto game. In order to win our example game, you have to pick the correct six numbers from 50 possible balls. The order in which the numbers are picked is not important; you just have to pick the correct six numbers.

The odds of picking a single correct number depend on how many balls have been chosen already. For instance, let's say none of the six numbers had been picked yet and you had to guess just one number correctly. Since there are 50 numbers to chose from, and since six balls are going to be picked, you have six tries at picking the number correctly. The odds of picking one number correctly are 50/6 = 8.33:1.

Using a similar calculation, we can determine the odds of picking another number correctly after one number has already been drawn. We know there are 49 balls left, and that five more balls will be drawn. So the odds of picking a number correctly after one has been drawn are 49/5 = 9.8:1.

Now let's say five numbers have been picked and you have to guess what the last number is going to be. There are only 45 balls left to choose from, but you only get one shot at it, so your odds are only 45:1.

In a similar manner, we can calculate the odds of picking the right number when two, three, four and five balls have been drawn. You know the odds of a coin toss resulting in heads are 1/2 = 2:1. The odds of two consecutive tosses both resulting in heads are 1/2 x 1/2 = 4:1. The odds of three consecutive tosses all resulting in heads are 1/2 x 1/2 x 1/2 = 8:1. The odds of picking all six lottery numbers are calculated the same way -- by multiplying together the odds of each individual event. In this case:

Some states have been increasing or decreasing the number of balls in order to change the odds. If the odds are too easy, then someone will win the jackpot almost every week and the prize will never grow. Large jackpots tend to drive more ticket sales. If the prize is not large enough, ticket sales can decrease. On the other hand, if the odds against winning are too great, ticket sales can also decline. It is important for each lottery to find the right balance between the odds and the number of people playing.

If you add just one number to our hypothetical lottery, so people now have to pick from 51 balls, the odds increase to 18,009,460:1.

Powerball
Some states have joined together to run multi-state Powerball lotteries. Since so many people can play, they need a game with really large odds against winning. In this multi-state Powerball lottery game, the winner has to pick the correct five numbers from a set of 50 balls, and they have to pick the single correct number from a separate set of 36 balls. So the odds of picking the correct number in this game are:

So let's say you pick the right six numbers and win a $10 million jackpot -- you're going to get $10 million, right? Well, sort of…