Should I buy all the 175.7M Mega Millions combinations?

In the Tuesday night Mega Millions lottery drawing on March 27, 2012, an estimated 170 to 240 million tickets were sold. We obtain these numbers by doubling the increase in the cash (from $255M to $341M) and annuity (from $356M to $240M) prizes from the 27th to the Friday March 30th drawing. (Note that while the Mega Millions annuity has been increased to $500M and cash to $359.4, we have chosen to use the original prize levels to estimate ticket sales).

However, for purposes of this discussion, we will assume that 200 million Mega Million tickets were sold for that drawing.

Since our ticket sale estimate is well above the total number of allowable Mega Millions combinations (175.7 million), one can logically ask:

Why was there no Mega Millions Jackpot winner?

The answer is: Duplicates!

Because of randomness, the percent of duplicate tickets increases rapidly as the number of tickets are sold. In Mega Millions, this increase begins to level off after sales exceed 400 million. As the illustration below shows, approximately 40% of the combinations printed are duplicates when 200 million individual tickets are sold.

Graph 1: Percent of Mega Millions Duplicates per Million Tickets Sold

This means that of the 200 million sold, 80 million are duplicates, and only 120 million are unique. If we divide the 120 million by the 175 million Mega Millions combinations, we find that approximately 68.6% of the combinations were generated. Subtracting 68.6% from 100%, we learn that there is a 31.4% chance that the number picked in the drawing will be a loser.

Graph 2: Chance of Losing Mega Millions per Million Tickets Sold

Given the fact that the new Mega Millions annuity jackpot is set at $476 million for Friday night, we can expect ticket sales to potentially reach or exceed 300 million. At this level, about 52% of these tickets will be duplicate combinations, and there will be an 18% chance that there will be no Jackpot winner at all.

So, getting to our title question:

Should I buy all the 175.7M Mega Millions combinations?

The answer depends on how you purchase them. If you purchase every single unique combination, then Yes, you are guaranteed to win the Jackpot. In this case, there will be no losing tickets. However, you have to worry about sharing the jackpot prize with others. Using Graph 1 above, we can say that at the 300 million ticket sales level, you have 50 - 50 chance of winning it alone or sharing it with one or more others.

If you take the Cash Option, you will lose money if you have to share it with 1 other person.

And, if you take the Annuity, you lose money if you share it with 2 others.

However, if you are lazy and just buy 175.7 million Quick Picks, you are not guaranteed to win the jackpot. As stated above, approximately 40% of these will be duplicates and you will have a 31.4% chance of losing.

The last thing to remember is that you will have to pay taxes on the money you receive. If you win the jackpot, you will be in the highest tax bracket possible, which means that you will be paying 28% to the federal government.Then, depending on the State where you purchased your winning ticket and live, you may have to pay state taxes as well.

So in summary, our advice is NO. You should not purchase the 175.7million Mega Millions combinations unless the jackpot becomes substantially higher to cover your taxes and the possible loss of sharing it with others.