Durango Bill's
Applied Mathematics


Mega Millions Odds

How to Calculate the Odds and Probabilities for the
Mega Millions Lottery


   Mega Millions odds and probabilities for the Mega Millions Jackpot – and how to calculate these Mega Millions odds. Additional calculations show tie probabilities and expected return on your “investment”.


Updated for the pick 5 out of 75, 1 out of 15 game.



Concise Table of Mega Millions Odds (Mathematical derivation below)


Ticket Matches       Payout           Odds               Probability
--------------------------------------------------------------------
5 White + Mega      Jackpot    1 in 258,890,850.00      0.000000003863
5 White No Mega   1,000,000    1 in 18,492,203.57       0.00000005408
4 White + Mega        5,000    1 in 739,688.14          0.000001352
4 White No Mega         500    1 in 52,834.87           0.00001893
3 White + Mega           50    1 in 10,720.12           0.00009328
3 White No Mega           5    1 in 765.72              0.001306
2 White + Mega            5    1 in 472.95              0.002114
1 White + Mega            2    1 in 56.47               0.01771
0 White + Mega            1    1 in 21.39               0.04675

Win something      Variable    1 in 14.71               0.0679916



Game Rules
   The numbers picked for the prizes consist of 5 numbers picked at random from a pool of 75 numbers (the White Numbers). Then a single number (the Mega Number) is picked from a second pool that has 15 numbers. If the results of these random number selections match one of the winning combinations on your lottery ticket, then you win something.


Combinatorics Calculations
   In any combinatorics problem where all possible outcomes are equally likely, the probability of a successful outcome is determined by finding the number of successful combinations, and then dividing by the total number of all combinations. There are nine possible configurations that will win something in the Mega Millions Lottery. For each of these, the probability of winning equals the number of winning combinations for that particular configuration divided by the total number of ways the Mega Millions numbers can be picked.


Mega Millions Total Combinations
   Since the total number of combinations for Mega Millions numbers is used in all the calculations, we will calculate it first. The number of ways 5 numbers can be randomly selected from a field of 75 is: COMBIN(75,5) = 17,259,390. (See the math notation page or Help in Microsoft's Excel for more information on “COMBIN”).

   For each of these 17,259,390 combinations there are COMBIN(15,1) = 15 different ways to pick the sixth number (the “Mega” number). The total number of ways to pick the 6 numbers is the product of these. Thus, the total number of equally likely Mega Millions combinations is 17,259,390 x 15 = 258,890,850. We will use this number for each of the following calculations.


Jackpot probability/odds (Payout varies)
   The number of ways the first 5 numbers on your lottery ticket can match the 5 White numbers is COMBIN(5,5) = 1. The number of ways your final number can match the Mega number is: COMBIN(1,1) = 1. The product of these is the number of ways you can win the Jackpot:  COMBIN(5,5) x COMBIN(1,1) = 1. The probability of success is thus: 1/258,890,850 = 0.0000000038626316844. If you express this as “One chance in ???”, you just divide “1” by the 0.0000000038626316844, which yields “One chance in 258,890,850”.

Match all 5 White numbers but not the Mega number (Payout = $1,000,000)
   The number of ways the 5 first numbers on your lottery ticket can match the 5 White numbers is COMBIN(5,5) = 1. The number of ways your final number can match any of the 14 losing Mega numbers is: COMBIN(14,1) = 14. (Pick any of the 14 losers.) Thus there are COMBIN(5,5) x COMBIN(14,1) = 14 possible combinations. The probability for winning $1,000,000 is thus 14/258,890,850 = .00000005407684358 or “One chance in 18,492,203.57”.

Match 4 out of 5 White numbers and match the Mega number (Payout = $5,000)
   The number of ways 4 of the 5 first numbers on your lottery ticket can match the 5 White numbers is COMBIN(5,4) = 5. The number of ways your fifth initial number can match any of the 70 losing White numbers is COMBIN(70,1) = 70.  The number of ways your final number can match the Mega number is: COMBIN(1,1) = 1. The product of these is the number of ways you can get this configuration:  COMBIN(5,4) x COMBIN(70,1) x COMBIN(1,1) = 350. The probability of success is thus: 350/258,890,850  = 0.00000135192109 or “One chance in 739,688.14”.

Match 4 out of 5 White numbers but not match the Mega number (Payout = $500)
   The number of ways 4 of the 5 first numbers on your lottery ticket can match the 5 White numbers is COMBIN(5,4) = 5. The number of ways your fifth initial number can match any of the 70 losing White numbers is COMBIN(70,1) = 70.  The number of ways your final number can match any of the 14 losing Mega numbers is: COMBIN(14,1) = 14. The product of these is the number of ways you can get this configuration:  COMBIN(5,4) x COMBIN(70,1) x COMBIN(14,1) = 4,900. The probability of success is thus: 4,900/258,890,850 = 0.000018926895 or “One chance in 52,834.87”.

Match 3 out of 5 White numbers and match the Mega number (Payout = $50)
   The number of ways 3 of the 5 first numbers on your lottery ticket can match the 5 White numbers is COMBIN(5,3) = 10. The number of ways the 2 losing initial numbers on your ticket can match any of the 70 losing White numbers is COMBIN(70,2) = 2,415.  The number of ways your final number can match the Mega number is: COMBIN(1,1) = 1. The product of these is the number of ways you can get this configuration:  COMBIN(5,3) x COMBIN(70,2) x COMBIN(1,1) = 24,150. The probability of success is thus: 24,150/258,890,850 = 0.000093282555 or One chance in 10,720.12”.

Match 3 out of 5 White numbers but not match the Mega number (Payout = $5)
   The number of ways 3 of the 5 first numbers on your lottery ticket can match the 5 White numbers is COMBIN(5,3) = 10. The number of ways the 2 losing initial numbers on your ticket can match any of the 70 losing White numbers is COMBIN(70,2) = 2,415.  The number of ways your final number can match any of the 14 losing Mega numbers is: COMBIN(14,1) = 14. The product of these is the number of ways you can get this configuration:  COMBIN(5,3) x COMBIN(70,2) x COMBIN(14,1) = 338,100. The probability of success is thus: 338,100/258,890,850 = 0.001306 or “One chance in 765.72”.

Match 2 out of 5 White numbers and match the Mega number (Payout = $5)
   The number of ways 2 of the 5 first numbers on your lottery ticket can match the 5 White numbers is COMBIN(5,2) = 10. The number of ways the 3 losing initial numbers on your ticket can match any of the 70 losing White numbers is COMBIN(70,3) = 54,740.  The number of ways your final number can match the Mega number is: COMBIN(1,1) = 1. The product of these is the number of ways you can get this configuration:  COMBIN(5,2) x COMBIN(70,3) x COMBIN(1,1) = 547,400. The probability of success is thus: 547,400/258,890,850  = 0.0021144 or “One chance in 472.95”.

Match 1 out of 5 White numbers and match the Mega number (Payout = $2)
   The number of ways 1 of the 5 first numbers on your lottery ticket can match the 5 White numbers is COMBIN(5,1) = 5. The number of ways the 4 losing initial numbers on your ticket can match any of the 70 losing White numbers is COMBIN(70,4) = 916,895.  The number of ways your final number can match the Mega number is: COMBIN(1,1) = 1. The product of these is the number of ways you can get this configuration:  COMBIN(5,1) x COMBIN(70,4) x COMBIN(1,1) = 4,584,475. The probability of success is thus: 4,584,475/258,890,850  = 0.017708 or “One chance in 56.47”.

Match 0 out of 5 White numbers and match the Mega number (Payout = $1)
   The number of ways 0 of the 5 first numbers on your lottery ticket can match the 5 White numbers is COMBIN(5,0) = 1. The number of ways the 5 losing initial numbers on your ticket can match any of the 70 losing White numbers is COMBIN(70,5) = 12,103,014.  The number of ways your final number can match the Mega number is: COMBIN(1,1) = 1. The product of these is the number of ways you can get this configuration:  COMBIN(5,0) x COMBIN(70,5) x COMBIN(1,1) = 12,103,014. The probability of success is thus: 12,103,014/258,890,850  = 0.04675 or “One chance in 21.39”.

Probability of winning something
   If we add all the ways you can win something we get:
1 + 14 + 350 + 4,900 + 24,150 + 338,100 + 547,400 + 4,584,475 + 12,103,014 = 17,602,404 different ways of winning something. If we divide by the 258,890,850, we get .0679916 as a probability of winning something.  1 divided by 0.0679916 yields “One chance in 14.71” of winning something.

Corollary
   You can get a close estimate for the number of tickets that were in play for any given game by multiplying the announced number of “winners” by the above 14.71. Thus, if the lottery officials proclaim that a given lottery drawing had 5 million “winners”, then there were about 5,000,000 x 14.71 ~= 73,500,000 tickets purchased that did not win the Jackpot. Alternately, there were about 73,500,000 - 5,000,000 ~= 68,500,000 tickets that did not win anything.

Note: This web page had over 50,000 hits for the large Jackpot on Jan. 4, 2011. If this is representative of what happened at Mega Millions headquarters, I extend my deepest sympathy to their computers.



Probability of multiple winning tickets (multiple winners) given “N” tickets in play

Note: All calculations assume that the numbers on any given ticket are picked randomly. In practice, many people pick numbers based on family birthdays, etc., and thus many tickets will have a preponderance of low numbers. As a consequence, the probabilities of a single Jackpot winner will be somewhat lower and the probabilities of no winner or multiple winners will tend to be slightly higher than the numbers shown below. Also if the numbers picked in the drawing are clustered at the high end of the 1-75 range, there will tend to be relatively less “partial match” winners. The reverse will hold true if the drawing numbers cluster in the low end of the number range.


Chart showing the probabilities of multiple winners

   The above chart shows the probabilities of “No Winners”, “One Winner”, and “Two or more Winners” for various numbers of tickets in play.

   Each entry in the following table shows the probability of “K” tickets holding the same winning Jackpot combination given that "N" tickets are in play for a given Mega Millions game. It is assumed that the number selections on each ticket are picked randomly. For example: If 100,000,000 tickets are in play for a Mega Millions game, then there is a 0.0507 probability that exactly two of these tickets will have the same winning combination.

Note: You can get a rough estimate of the number of tickets in play as follows. If the preceding Mega Millions game had no Jackpot winner, multiply the dollar increase in the Jackpot from the preceding game to the current game by 3. (Use the cash payout amount). For example, if the preceding game had a cash payout amount of $50,000,000 and the current game has a cash payout amount of $70,000,000, then there are about 3 x (70,000,000 – 50,000,000) = 60,000,000 tickets in play for the current game. (Past Jackpot amounts can be seen at: http://www.lottoreport.com/mmsales.htm. The cash payout value for these would be about one-half the announced Jackpot amount.)

“N”   Number                           “K”
of tickets         Number of tickets holding the Jackpot combination
in play            0       1       2       3       4       5       6
----------------------------------------------------------------------
20,000,000      0.9257  0.0715  0.0028  0.0001  0.0000  0.0000  0.0000
40,000,000      0.8568  0.1324  0.0102  0.0005  0.0000  0.0000  0.0000
60,000,000      0.7931  0.1838  0.0213  0.0016  0.0001  0.0000  0.0000
80,000,000      0.7342  0.2269  0.0351  0.0036  0.0003  0.0000  0.0000
100,000,000     0.6796  0.2625  0.0507  0.0065  0.0006  0.0000  0.0000
120,000,000     0.6291  0.2916  0.0676  0.0104  0.0012  0.0001  0.0000
140,000,000     0.5823  0.3149  0.0851  0.0153  0.0021  0.0002  0.0000
160,000,000     0.5390  0.3331  0.1029  0.0212  0.0033  0.0004  0.0000
180,000,000     0.4989  0.3469  0.1206  0.0279  0.0049  0.0007  0.0001
200,000,000     0.4618  0.3568  0.1378  0.0355  0.0069  0.0011  0.0001
220,000,000     0.4275  0.3633  0.1544  0.0437  0.0093  0.0016  0.0002
240,000,000     0.3957  0.3669  0.1700  0.0525  0.0122  0.0023  0.0003
260,000,000     0.3663  0.3679  0.1847  0.0618  0.0155  0.0031  0.0005
280,000,000     0.3391  0.3667  0.1983  0.0715  0.0193  0.0042  0.0008
300,000,000     0.3139  0.3637  0.2107  0.0814  0.0236  0.0055  0.0011
320,000,000     0.2905  0.3591  0.2219  0.0914  0.0283  0.0070  0.0014
340,000,000     0.2689  0.3532  0.2319  0.1015  0.0333  0.0088  0.0019
360,000,000     0.2489  0.3462  0.2407  0.1116  0.0388  0.0108  0.0025
380,000,000     0.2304  0.3382  0.2482  0.1214  0.0446  0.0131  0.0032
400,000,000     0.2133  0.3296  0.2546  0.1311  0.0506  0.0157  0.0040
420,000,000     0.1974  0.3203  0.2598  0.1405  0.0570  0.0185  0.0050
440,000,000     0.1828  0.3106  0.2640  0.1495  0.0635  0.0216  0.0061
460,000,000     0.1692  0.3006  0.2671  0.1582  0.0703  0.0250  0.0074
480,000,000     0.1566  0.2903  0.2692  0.1663  0.0771  0.0286  0.0088
500,000,000     0.1450  0.2800  0.2703  0.1740  0.0840  0.0325  0.0104


Any entry in the table can be calculated using the following equation:

Prob. = COMBIN(N,K) x (Pwin^K) x (Pnotwin^(N-K))

Where:
N = Number of tickets in play
K = Number of tickets holding the Jackpot combination
Pwin = Probability that a random ticket will win ( = 1 / 258,890,850  =  0.00000000386)
Pnotwin = (1.0 - Pwin)  = 0.99999999614
COMBIN(N,K)  = number of ways to select K items from a group of N items
x   =  multiply terms
^   =  raise to power (e.g.  2^3 = 8 )


Sample Calculation to Find the Expected Shared Jackpot Amount
When a Large Number of Tickets are in Play

   For this example we will assume the cash value of the Jackpot is $120,000,000 (Cash Value) and there are 100,000,000 tickets in play for the current game. Probability values are from the “100,000,000” row above.

Number of                     Jackpot paid       Contribution
winners      Probability     to each winner     (Col 2 x Col 3)
“--------------------------------------------------------------”
0              .6796                     0                  0
1              .2625           120,000,000         31,500,000
2              .0507            60,000,000          3,042,000
3              .0065            40,000,000            260,000
4              .0006            30,000,000             18,000
Total                                              34,820,000

   This Total then has to be divided by 1 - .6796 = .3204 to give a weighted Jackpot amount of  34,820,000 / .3204 = $108,676,654 which would be used as the payout amount figure used in the “Return on Investment” section below.

   These calculations can be used to form an index showing how much the quoted amount of the Jackpot should be reduced to allow for the expected number of co-winners. In the table below, for any given number of tickets in play, the quoted Jackpot should be multiplied by the value in the next column to produce the true expected value of a winning ticket. For example, if there are 200,000,000 tickets in play for a quoted $300,000,000 Jackpot, then the expected value for the Jackpot becomes $300,000,000 x  0.8166 = $244,980,000 to adjust for the possibility that a winning ticket will have to split the Jackpot with some other winning ticket.


  Number of       Mult. Jackpot by       Number of     Mult. Jackpot by
  Tickets         this ratio for         Tickets       this ratio for
  in play         possible sharing       in play       possible sharing
          0            1.0000          200,000,000          0.8166
 20,000,000            0.9808          220,000,000          0.7995
 40,000,000            0.9617          240,000,000          0.7827
 60,000,000            0.9428          260,000,000          0.7661
 80,000,000            0.9242          280,000,000          0.7497
100,000,000            0.9057          300,000,000          0.7337
120,000,000            0.8874          320,000,000          0.7179
140,000,000            0.8694          340,000,000          0.7023
160,000,000            0.8516          360,000,000          0.6871
180,000,000            0.8340          380,000,000          0.6721
200,000,000            0.8166          400,000,000          0.6574




Return on Investment

Finally, it is interesting to calculate what the long term expected return is for each $1.00 lottery ticket that you buy

   The first task is to construct a table where each row lists the winning combination, the payout, the probability of this payout, and the contribution to the expected return (Equals payout times probability.) The probabilities are the same ones we derived earlier. An $85,000,000 cash payout (decline the annuity) is assumed for the Jackpot. (Could be your portion of a shared Jackpot.)

Combination         Payout     Probability   Contribution
---------------------------------------------------------
5 White + Mega   $85,000,000   3.86263E-09     $0.3283 
5 White No Mega    1,000,000   5.40768E-08      0.0541
4 White + Mega         5,000   1.35192E-06      0.0068
4 White No Mega          500   1.89269E-05      0.0095
3 White + Mega            50   9.32826E-05      0.0047
3 White No Mega            5   0.001305956      0.0065
2 White + Mega             5   0.002114405      0.0106
1 White + Mega             2   0.017708138      0.0354
0 White + Mega             1   0.046749485      0.0467

Total                          0.025069987      0.5026

Thus, for each $1.00 that you spend for Mega Millions tickets, you can expect to get back about $0.50. Of course you get to pay taxes on any large payout, so your net return is even less.



Expected after tax return on your $1.00 ticket investment when a large Jackpot is in play

  While the above calculation represents an average Mega Millions game, we might ask what the expected after tax return on your investment might be if a large Jackpot exists. Since there are no precedents for large Jackpots for the current game, we will make an estimate of what might occur.

We will use the following assumptions:
Quoted Jackpot: $700 million (Sum of annuity payments over 30 years)
Cash Jackpot: $400 million (Take the money now – used/valid for calculations)
Cash Jackpot after 40% discount for taxes: $240 million (Used for calculations)
Number of tickets in play: 600,000,000


   The ratio for 600,000,000 tickets in play is 0.525974338. (If there is a winner, the Jackpot will probably split 1/0.525974338 ~= 2 ways.) Thus the expected before tax cash value of the Jackpot would be $400 million x 0.525974338 = $210,389,736. This has to be reduced by 40% for taxes, so your after tax expected amount if you win is only $126,233,841.

   The next 2 prize amounts must also be reduced by 40% for taxes. Thus the $1,000,000 2nd prize becomes $600,000 and the $5,000 3rd prize becomes $3,000. It is assumed that all other prizes are not reported for income tax purposes.

We can now construct a table where the:
“Amount” is the before tax amount if your ticket wins something
“Probability” equals the probability of wining this combination
“Before tax Exp. Return” equals “Amount” times “Probability” 
“After tax Exp. Return” equals the above less 40% taxes for large prizes

                                               Before tax     After tax
Ticket              Amount     Probability    Exp. Return   Exp. Return
5 Wh + Mega   $210,389,736     3.86263E-09       $0.8127       $0.4876
5 Wh no Mega     1,000,000     5.40768E-08        0.0541        0.0324
4 Wh + Mega          5,000     1.35192E-06        0.0068        0.0041
4 Wh no Mega           500     1.89269E-05        0.0095        0.0095
3 Wh + Mega             50     9.32826E-05        0.0047        0.0047
3 Wh no Mega             5     0.001305956        0.0065        0.0065
2 Wh + Mega              5     0.002114405        0.0106        0.0106
1 Wh + Mega              2     0.017708138        0.0354        0.0354
0 Wh + Mega              1     0.046749485        0.0467        0.0467
Totals                                            0.9869        0.6375

   The above table shows us that in this hypothetical large Jackpot setting, your expected before tax return is only $0.9869 for each $1.00 ticket that you buy, and your after tax return is only $0.64 for each $1.00 ticket that you buy.


Percentile Expected Returns on Ticket Purchases

   The average return per $1.00 ticket includes the extremely low probability that you might win a large prize – for example $5,000 or more. As a practical matter, it is unlikely that you will ever buy enough tickets (fork out enough money) to ever have much of a chance for any of the large prizes. Thus it is probable that all you will ever get back from your ticket purchases are piddling small amounts.

    The percentages for these small amounts can be calculated. The table below shows the percentage chances for various “piddling returns”.

If you spend $1,000 to buy 1,000 tickets (1 ticket for each of 1,000 Mega Millions games) there is a:
48.97 % chance that you will get back $100 or less
60.93 % chance that you will get back $105 or less
70.86 % chance that you will get back $110 or less
80.52 % chance that you will get back $117 or less
89.95 % chance that you will get back $135 or less
94.95 % chance that you will get back $155 or less
98.00 % chance that you will get back $567 or less
99.01 % chance that you will get back $602 or less
99.49 % chance that you will get back $616 or less
99.80 % chance that you will get back $659 or less

Even if you buy 1,000 tickets, your chance of winning a $5,000 or larger prize is less than 0.2 %.




Megaplier

Some states use a Megaplier feature to increase non-jackpot prizes by 2, 3, 4 or 5 times; it costs an additional $1 per play.

   If your state has a “Megaplier” and if your state follows the probabilities posted on the Mega Million web site, then a calculation can be made for the expected return if you pay an additional $1.00 to participate in the Megaplier play. To find out the expected return, we construct a table to calculate the average expected multiplier.


Multiple     Odds        Probability      Contribution
   2       1 in 7.5        0.133333         0.266667
   3       1 in 3.75       0.266667         0.800000
   4       1 in 5          0.200000         0.800000
   5       1 in 2.5        0.400000         2.000000
Totals                     1.000000         3.866667

Odds are from the Mega Millions web page: http://www.megamillions.com/how-to-play
Probability = 1 divided by the odds
Contribution = “Multiple” times “Probability”

   1.0 has to be subtracted from this 3.866667 because you would win “1 unit” of the sub prizes just from your simple ticket purchase. This leaves a “bonus contribution multiplier” of just 2.866667.

  Thus we have calculated that the average multiplier is 2.866667. We then multiply the average extra expected return for all the sub-prizes (previously calculated) by this 2.86667 to get the expected return if you buy the “Megaplier” option.

(0.0541 + 0.0068 + 0.0095 + 0.0047 + 0.0065 + 0.0106 + 0.0354 + 0.0467) x 2.866667 = $0.50

Thus if you pay another $1.00 to buy the Megaplier option, your expected before tax return is $0.50.



2nd Thoughts
There are about 1.5 automobile caused fatalities for every 100,000,000 vehicle-miles. (2000 to 2005 average data http://en.wikipedia.org/wiki/Transportation_safety_in_the_United_States ) If you drive one mile to the store to buy your Mega Millions ticket and then return home, you have driven two miles. Thus the probability that you will join this statistical group is 2 x 1.5 / 100,000,000 = 0.00000003. This can also be stated as “One in 33,333,333”. Thus, if you drive to the store to buy your Mega Million ticket, your chance of being killed (or killing someone else) is nearly 8 times greater than the chance that you will win the Mega Millions Jackpot.

   Alternately, if you “played” Russian Roulette 100 times per day, every day for 71 years, with Mega Millions Jackpot odds, you would have better than a 99% chance of surviving.


3rd Thoughts
A lottery is a “Zero-sum game”. What one group of participants gains in cash, the other group of participants must lose. If we made a list of all the participants in a lottery, it might include:

1) Federal Government (Lottery winnings are taxable)
2) State Governments (Again lottery winnings are taxable)
3) State Governments (Direct share of lottery ticket sales)
4) Merchants that sell tickets (Paid by the lottery organizers)
5) Lottery companies (Hint: They are not doing all this for free)
6) Advertisers and promoters (Paid by the lottery companies)
7) Lottery ticket buyers (Buy lottery tickets and receive payouts)

The winners in the above list are:
1) Federal Government
2) State Government (Taxes)
3) State Government (Direct share)
4) Merchants that sell tickets
5) Lottery companies
6) Advertisers and promoters

And the losers are:
    (Mathematically challenged and proud of it)

Also please see the related calculations for Powerball.


 


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