Friday, March 30, 2012

Probability of a Mega Millions rollover or having a single winner decrease as sales approach 600M

As tonight's record Mega Millions drawing approaches, ticket sales are going off the charts. We originally expected that 400 million tickets would be sold, but with the recent increases in the jackpot prize from $476M to $540M $640M, we now believe that up to 600 million tickets may be sold.

Thus, we asked ourselves two questions:
  1. What is the probability that there will be no winner in tonight's drawing?
  2. What is the probability that there will only be one sole winner?
Using our brute force duplicate ticket estimator, we have produced two tables for your reference.

The first summarizes the probabilities of: 0 to 8 individual winning tickets. As shown, the probability of not having a winner tonight and rolling over again drops from 18% to 3% as ticket sales increase from 300 million to 600 million. Similarly,  we can see that the probability of having a single winner decreases from 39% to 27%, while that of two winners decreases slightly from 28% to 26%. Interestingly, we find the probabilities of 3 to 5 winners rapidly increase as sales approach 600 million. We believe the likelihood that 6 or more winning combinations will be sold is rather slim.

Table 1: Probability of Mega Millions Winning Tickets
Winning Tickets 300M Sold  400M Sold 500M Sold 600M Sold
None, No winner 18.1% 10.3% 5.8% 3.3%
1 Winner Only 39.3% 35.4% 31.2% 27.4%
2 Winner Only 28.2% 29.5% 28.2% 25.9%
3 Winner 12.4% 18.4% 21.5% 22.2%
4 Winner 1.9% 5.9% 11.0% 14.9%
5 Winner
0.5% 2.3% 5.6%
6 Winner

0.1% 0.6%
7 Winner

8 Winner

Total Probability 100.0% 100.0% 100.0% 100.0%

The second table summarizes the distribution of the tickets sold. It estimates the absolute number of tickets in each category. Here we find that approximately 31.9 million combinations will not be generated when sales are at the 300M level, but only 5.8 million combinations will be unsold when 600M tickets are printed. In all other cases, the absolute number of tickets per category increase as sales increase. Interestingly enough, we can see how the size of the 6 and 7 tuplets begins to grow.

Table 2: Distribution of Mega Millions Tickets Sold
Numbers 300M Sold  400M Sold500M Sold600M Sold
All Combinations 175,711,536 175,711,536 175,711,536 175,711,536
Unsold 31,864,733 18,036,279 10,209,010 5,778,566
1 Only (Singles) 143,846,803 157,675,257 165,502,526 169,932,970
2  (Doubles) 103,460,042 131,466,722 149,526,982 160,511,932
3 (Triples) 45,526,185 82,214,057 114,388,867 137,818,542
4 (Quadruples) 7,022,773 26,431,098 58,127,495 92,689,034
5 (Quintuples) 144,138 2,198,990 12,023,012 35,013,149
6 (Sextuplets) 59 13,876 430,590 3,988,410
7 (Septuplets) 0 0 528 45,963
8 (Octuplets)

Sum 300,000,000 400,000,000 500,000,000 600,000,000

We wish everyone the best of luck tonight and hope your dreams come true. But, as we can see from the above, remember to set your expectations according to the probability that you may have to share the jackpot prize with somebody else.



Wednesday, March 28, 2012

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.

Wednesday, March 21, 2012

Multiple Lottery Ticket Odds Calculator Released

With all the excitement of winning the large Mega Millions and Powerball jackpots recently, we realized that many players were searching for a calculator that would show the odds (or chances) of winning when multiple tickets are purchased.

In response, we created and released a new Multi-Ticket Lottery Odds Calculator gadget which can be found on the right hand sidebar of this blog (about 1/4 down the page), on our Lottery Power Picks News blog, and in the Google gadgets directory.  To use this calculator, simply enter the number of tickets that you purchased and press the enter (or return) key. The odds of winning the top 2 prizes will automatically be displayed.  In our rush to release this gadget, we sacrificed making this a "pretty" display.

You can either use the calculator directly on this blog, or you can add it to your own iGoogle home page  by clicking on the "+ Google" button directly under the calculator.

Let us know what you think. 

Thanks, JL ......
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