Please bear with me; I am going to post a lengthy description of odds calculations for multiple ticket purchases. Please follow the logic, and don't get frustrated.
Some folks here are having a difficult time with the mathematics of odds calculations. My guess is that dealing with huge numbers is causing a problem for them.
Maybe if we talk about small numbers, things will become clear.
After all, the subject of mathematics works the same way no matter how big or small the numbers are.
For example, 1 + 1 = 2, and 10 + 10 = 20, and 1,000,000 + 1,000,000 = 2,000,000. The same concept works, no matter how big the numbers are. It works the same for addition, subtraction, multiplication, and division.
A Small Example
Now then, let's take our odds calculation into the realm of "tens" instead of "millions" of calculations, and then we'll ramp up the discussion back to millions once we have the concepts down.
Let's say our lottery has a total of 10 numbers being selected from a drum, and you buy one ticket. Your odds of winning on that ticket are 1 in 10. I think everyone can agree with that statement.
It is the same as saying you have covered one-tenth of the possible combinations. In other words, an odds statement is a fraction — "one over 10", or 1/10. You can do the division (1 divided by 10) to say I have a "0.1" chance of winning. And as we also know, decimals can be expressed as percentages by simply moving the decimal place two positions to the right. So our "0.1" chance of winning becomes 10%.
That is all easy to understand when dealing with small numbers, especially tens. We have expressed our odds of winning in four different ways:
- 1 in 10 (a traditional odds statement)
- 1/10 (a fraction)
- 0.1 (a decimal)
- 10% (a percentage — we have covered 10% of the possible combinations)
All of those four ways should be easily and quickly agreed upon by all. The concept of 1 in 10 chances is very easy to understand and express.
Buying More Tickets
OK, so we buy 2 tickets for the drawing. Let's express those two tickets in the same four ways as we did for our 1-ticket purchase:
- 2 in 10
- 20% — we have covered 20% of the possible combinations
Just to be clear: We do NOT think of our lottery drawing as two separate "1 in 10" odds, because we would have to conduct two separate drawings for that to be true. We are still only conducting one drawing, but now two combinations are covered out of the possible 10 combinations.
Another way to look at it is visually. You can see how our two purchased numbers represents 20% of the possible combinations, and the ones we have NOT purchased is 80% of the combinations. Our math checks out perfectly.
Expressing as "1 in ____"
One of the points I made previously was that all odds calculations can be expressed as "1 in _____", meaning you have "one chance in _______ (some number)" to win. We all know it's possible to express odds that way, because that is really the only way the lottery EVER speaks about odds, no matter how complicated the game is to play.
A great example of this is the Canada Lotto Max game. For $5, you get a ticket with 3 separate lines (combinations) on it. You get to pick your own numbers for the first line, but the second two lines are always quick picks. Regardless of the way the numbers are selected, each $5 purchase gets you the equivalent of buying three tickets. So how does the lottery express the odds for that 3-ticket purchase for $5? As "1 in _____" odds. They don't say "you have three separate chances of "1 in ____", they combine the three combinations and accurately tell you what your "1 in ____" odds of winning are. You can see it for yourself in the Prizes section at http://www.olg.ca/lotteries/games/howtoplay.do?game=lottomax.
Getting back to my little lottery drawing example here, we can see that when I bought a second ticket above, I expressed the odds as "2 in 10". So what is the "1 in _____" become?
That's pretty easy to calculate. Since one of the four ways to express the odds is a fraction, we can just simplify the fraction — something taught in elementary mathematics.
Simplifying our 2/10 fraction above, we divide the top and bottom of the fraction by the lowest common denominator, which is 2, and our new fraction becomes 1/5.
Now our four methods of expressing the 2-ticket purchase become:
Notice something here! Although we have changed the first two expressions, the last two have stayed the same.
How is that possible? Because reducing a fraction does not change its value, it only changes the expression of it. 1/5 is the same number as 2/10. They both equate to 20%.
So hopefully the skeptics here will see that there is no "magic fraction reduction", or anything strange going on. We have not made the game somehow easier to win — all we have done is accurately reflect the mathematical way of stating of chances of winning.
When expressed visually, you can see how saying "1 in 5" instead of "2 in 10" does not magically make the game easier to win. It is still a 20% chance of winning.
Chances of not purchased
Now, let's buy 5 tickets for the drawing. The numbers become:
- 1 in 2 (or, 5 in 10 — same thing)
- 1/2 (or, 5/10 — same thing)
Visually, it's easy to see this is correct:
Scaling the Concept to Mega Millions
There is nothing magical about calculating the odds of multiple ticket purchases for Mega Millions. The same exact mathematical calculations are used, no matter how many combinations there are.
To review a 1-ticket purchase of Mega Millions, our four ways of expressing the odds are:
- 1 in 175,711,536
Wow, that's a pretty small chance of winning. Look at that percentage!
So let's use the example discussed in this thread of purchasing 13 tickets. Here's the new odds:
- 1 in 13,516,272 (same thing as 13 in 175,711,536)
- 1/13,516,272 (same thing as 13/175,711,536)
If this is where I lose you, then let's step back to the discussion above where I mentioned the Canada Lotto Max game. In that game you get 3 chances (tickets) per purchase, and the lottery expresses the odds as a "1 in ____" number. So let's see what the Mega Millions odds would be with a 3-ticket purchase.
Because Mega Millions has a larger number matrix than Lotto Max, we know that the odds must be steeper to win.
In other words, the Ontario Lottery has published the Lotto Max odds of winning the jackpot as 1 in 28,633,528, so our 3-ticket Mega Millions purchase, if calculated correctly, will have WORSE odds than that.
And here is the Mega Millions calculation for 3 tickets:
- 1 in 58,570,512 (same as 3 in 175,711,536)
- 1/58,570,512 (same as 3/175,711,536)
Well, we can see that our 1 in 58,570,512 chances are definitely worse than Lotto Max's 1 in 28,633,528 chances of winning, when the playing field is leveled by purchasing 3 Mega Millions tickets.
In fact, we would have to purchase 7 Mega Millions tickets in order to achieve approximately the same odds of winning Lotto Max (with the Lotto Max $5 purchase):
- 1 in 25,101,648 (same as 7 in 175,711,536)
- 1/25,101,648 (same as 7/175,711,536)
I really hope this sheds some light on the discussion of odds calculations for multiple ticket purchases.
I do not intend this information to reinforce some notion that buying multiple tickets somehow make the game much easier to win. In fact, if you think buying 13 tickets and making the odds 1 in 13,516,272 is "much easier to win", then you are deluding yourself. The odds against you are still astronomical either way.
However, one thing I am doing here is being FACTUAL. It IS possible to make your odds BETTER by buying more than one ticket. The mathematics I have described here accurately reflect exactly how much better your odds become. Frankly, to those who do not like the way it is being expressed, then it would be best for you to not think in terms of odds at all, because it will only become upsetting for you. Mathematics is a "black and white" subject. There are not two correct answers here, only the one answer I am giving you.
On the other hand, if you have always been confused by odds for multiple tickets, and this helps clarify the topic, then it is my pleasure to have helped!
Good luck in tonight's big drawing!