No matter how many people play Mega Millions, the chances of winning never change. The chances of winning are based on the amount of number combinations, not the number of players. Quick Pick tickets are generated randomly by the terminal at the retailer, and there is no central computer that is controlling ticket generation. The chances of buying a Mega Millions ticket with the same numbers as someone else are the same as the chances of winning the jackpot -- about 1 in 175 million.
Many people ask us to explain how their chances of winning are determined. So, let's start at the beginning!
The Mega Millions game is played by selecting five numbers out of 56 choices (the numbers from 1 to 56), and by selecting one extra number from 1 to 46 (that's the Megaball®). When it's time to draw the winning numbers, there are two machines tumbling the ball sets at the same time: one machine has the 56 white balls, and the other one has the 46 gold "Megaballs®." So the Mega Millions game is like holding two drawings at the same time; two independent events played simultaneously. Any chances of winning has to consider what happens to the number selections within one drum combined with the possibilities of what's happening in the other drum. That idea is really important to keep in mind. A winning set of numbers will be whatever five balls are selected out of the white-ball drum, plus the one ball selected from the gold-ball drum.
But the chances of winning look confusing because even at the smallest winning level (just getting the Megaball® right), many people think the chances of winning that should be 1 in 46 (one correct choice out of a drum of 46 balls, right?). But we're not just playing that one drum; in order to say you "only matched the Megaball®" that means you have to have missed all five of the white balls that were chosen in that other drum! The chances of doing that combined with your 1-in-46 shot of getting the Megaball® actually come out to being about 1 in 75. In other words, there is about one chance in 75 that you'd pick the winning Megaball® number and miss selecting any of the five winning white-ball numbers.
So when you hear a "chances of winning" calculation, it's describing the chance that you will have chosen the winning numbers, including the chances you chose numbers that didn't get drawn. Since there are several ways to win (from zero to all five of the white balls, with and without having the Megaball®), that means people who calculate the chances of winning have to figure each level of winning or not winning in each of the two drums.
For example, in order to win the big jackpot, a winner must have chosen all five white balls correctly, as well as the Megaball® number. To describe the chances of winning, you have to say what your possible chances are of choosing the winning balls against choosing any of the 51 non-winning balls in the one drum, combined with your chances of choosing the winning Mega Millions against choosing any of the 45 non-winning " Megaballs®"in that drum. That's a lot of combinations - more than 175 million, in fact. But there's an easy formula that gets you there.
The mathematical formula looks like this:
Each part of that formula, which is in parentheses, is a "binomial coefficient," which is a calculation on how many ways there are to make different combinations. In our case, we're talking about how many combinations of lottery balls you could make with a set of 56 (or 46) balls. What that means is that for whatever numbers you have, the binomial can be rewritten as follows:
X! means "x factorial"; for example, 5! is 5 x 4 x 3 x 2 x 1 = 120.
Now we're ready to look at winning that jackpot!
To do that, we said you'd have to describe the chances of picking five numbers including 51 non-winning ones out of a total of 56. Then you combine that with the figure to describe your chances of picking one winning Megaball® number, including choosing 45 non-winning numbers. So using our binomials and factorials, it looks like this:
So, 56 x 55 x 54 x 53 x 52... = 458,377,920 and 5 x 4 x 3 x 2 x 1 = 120; divide one by the other:
458,377,920 / 120 = 3,819,816 (chances of 3,819,816 to 1 of picking all 5 white balls out of 56)
And don't forget that now we have to combine the possibilities of choosing the Megaball®, which we know is one in 46. That formula works just like the one above:
So, 3,819,816 / 1 x 46 / 1 = 175,711,536.
Now, that's just one win scenario: the big jackpot. But you can also win eight other prizes ranging from $2 to $250,000.
(In order to win a prize, your Megaball® number must at least match the Megaball® number drawn, or you must match at least three of the five white-ball numbers drawn.)
When you combine the chances of all those levels and all those combinations, it comes out to roughly 1 in 40.
So each player has a 1-in-40 chance to win any of the prizes (jackpot, $250,000, $10,000, $150, $10, $7, $3, or $2)!
We hope this helps to better explain the chances of winning the Mega Millions jackpot.