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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)


0
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.

Cheers,

JL...............

1 comment:

  1. It would be great if the lottery would publish the multiplet distribution after every draw.

    ReplyDelete

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