Durango Bill's
Applied Mathematics
Mega Millions Odds
How to Calculate the Odds and Probabilities for the
Mega Millions Lottery
Applied Mathematics
Mega Millions Odds
How to Calculate the Odds and Probabilities for the
Mega Millions Lottery
If the only thing you are interested in is the probability (odds) of
winning the Mega Millions Jackpot, the Mega Millions Lottery gives a
concise table at their web site. We will give the same information here, but
also show you how these odds are calculated.
Game Rules
The numbers picked for the prizes consist of 5 numbers picked at random from a pool of 56 numbers (the White Numbers). Then a single number (the Mega Number) is picked from a second pool that has 46 numbers. If the results of these random number selections match one of the winning combinations on your lottery ticket, then you win something.
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 56 is: COMBIN(56,5) = 3,819,816. (See the math notation page or Help in Microsoft's Excel for more information on “COMBIN”).
For each of these 3,819,816 combinations there are COMBIN(46,1) = 46 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 3,819,816 x 46 = 175,711,536. 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: 1/175,711,536 = 0.00000000569114597006. If you express this as “One chance in ???”, you just divide “1” by the 0.00000000569114597006, which yields “One chance in 175,711,536”.
Match all 5 White numbers but not the Mega number (Payout = $250,000)
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 45 losing Mega numbers is: COMBIN(45,1) = 45. (Pick any of the 45 losers.) Thus there are COMBIN(5,5) x COMBIN(45,1) = 45 possible combinations. The probability for winning $250,000 is thus 45/175,711,536 = 0.000000256101568653 or “One chance in 3,904,700.80”.
Match 4 out of 5 White numbers and match the Mega number (Payout = $10,000)
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 51 losing White numbers is COMBIN(51,1) = 51. 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 this configuration: COMBIN(5,4) x COMBIN(51,1) x COMBIN(1,1) = 255. The probability of success is thus: 255/175,711,536 = 0.00000145124222237 or “One chance in 689,064.85”.
Match 4 out of 5 White numbers but not match the Mega number (Payout = $150)
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 51 losing White numbers is COMBIN(51,1) = 51. The number of ways your final number can match any of the 45 losing Mega numbers is: COMBIN(45,1) = 45. The product of these is the number of ways you can win this configuration: COMBIN(5,4) x COMBIN(51,1) x COMBIN(45,1) = 11,475. The probability of success is thus: 11,475/175,711,536 = 0.0000653059000065 or “One chance in 15,312.55”.
Match 3 out of 5 White numbers and match the Mega number (Payout = $150)
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 51 losing White numbers is COMBIN(51,2) = 1,275. 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 this configuration: COMBIN(5,3) x COMBIN(51,2) x COMBIN(1,1) = 12,750. The probability of success is thus: 12,750/175,711,536 = 0.0000725621111183 or “One chance in 13,781.30”.
Match 3 out of 5 White numbers but not match the Mega number (Payout = $7)
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 51 losing White numbers is COMBIN(51,2) = 1,275. The number of ways your final number can match any of the 45 losing Mega numbers is: COMBIN(45,1) = 45. The product of these is the number of ways you can win this configuration: COMBIN(5,3) x COMBIN(51,2) x COMBIN(45,1) = 573,750. The probability of success is thus: 573,750/175,711,536 = 0.00326529500032 or “One chance in 306.25”.
Match 2 out of 5 White numbers and match the Mega number (Payout = $10)
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 51 losing White numbers is COMBIN(51,3) = 20,825. 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 this configuration: COMBIN(5,2) x COMBIN(51,3) x COMBIN(1,1) = 208,250. The probability of success is thus: 208,250/175,711,536 = 0.00118518114827 or “One chance in 843.75”.
Match 1 out of 5 White numbers and match the Mega number (Payout = $3)
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 51 losing White numbers is COMBIN(51,4) = 249,900. 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 this configuration: COMBIN(5,1) x COMBIN(51,4) x COMBIN(1,1) = 1,249,500. The probability of success is thus: 1,249,500/175,711,536 = 0.00711108688959 or “One chance in 140.63”.
Match 0 out of 5 White numbers and match the Mega number (Payout = $2)
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 51 losing White numbers is COMBIN(51,5) = 2,349,060. 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 this configuration: COMBIN(5,0) x COMBIN(51,5) x COMBIN(1,1) = 2,349,060. The probability of success is thus: 2,349,060/175,711,536 = 0.0133688433524 or “One chance in 74.80”.
Probability of winning something
If we add all the ways you can win something we get:
1 + 45 + 255 + 11,475 + 12,750 + 573,750 + 208,250 + 1,249,500 + 2,349,060 = 4,405,086 different ways of winning something. If we divide by the 175,711,536, we get 0.0250699874367 as a probability of winning something. 1 divided by 0.0250699874367 yields “One chance in 39.89” 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 39.89. Thus, if the lottery officials proclaim that a given lottery drawing had 2 million “winners”, then there were about 2,000,000 x 39.89 ~= 79,780,000 tickets purchased that did not win the Jackpot. Alternately, there were about 79,780,000 - 2,000,000 ~= 77,780,000 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.
(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 1-56 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.)
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 0.0917 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 7 8
-------------------------------------------------------------------------------------
20,000,000 0.8924 0.1016 0.0058 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000
40,000,000 0.7964 0.1813 0.0206 0.0016 0.0001 0.0000 0.0000 0.0000 0.0000
60,000,000 0.7107 0.2427 0.0414 0.0047 0.0004 0.0000 0.0000 0.0000 0.0000
80,000,000 0.6343 0.2888 0.0657 0.0100 0.0011 0.0001 0.0000 0.0000 0.0000
100,000,000 0.5660 0.3221 0.0917 0.0174 0.0025 0.0003 0.0000 0.0000 0.0000
120,000,000 0.5051 0.3450 0.1178 0.0268 0.0046 0.0006 0.0001 0.0000 0.0000
140,000,000 0.4508 0.3592 0.1431 0.0380 0.0076 0.0012 0.0002 0.0000 0.0000
160,000,000 0.4023 0.3663 0.1668 0.0506 0.0115 0.0021 0.0003 0.0000 0.0000
180,000,000 0.3590 0.3678 0.1884 0.0643 0.0165 0.0034 0.0006 0.0001 0.0000
200,000,000 0.3204 0.3647 0.2075 0.0787 0.0224 0.0051 0.0010 0.0002 0.0000
220,000,000 0.2859 0.3580 0.2241 0.0935 0.0293 0.0073 0.0015 0.0003 0.0000
240,000,000 0.2552 0.3485 0.2380 0.1084 0.0370 0.0101 0.0023 0.0004 0.0001
260,000,000 0.2277 0.3369 0.2493 0.1230 0.0455 0.0135 0.0033 0.0007 0.0001
280,000,000 0.2032 0.3238 0.2580 0.1370 0.0546 0.0174 0.0046 0.0011 0.0002
300,000,000 0.1813 0.3096 0.2643 0.1504 0.0642 0.0219 0.0062 0.0015 0.0003
320,000,000 0.1618 0.2947 0.2684 0.1629 0.0742 0.0270 0.0082 0.0021 0.0005
340,000,000 0.1444 0.2795 0.2704 0.1744 0.0844 0.0326 0.0105 0.0029 0.0007
360,000,000 0.1289 0.2641 0.2705 0.1847 0.0946 0.0388 0.0132 0.0039 0.0010
380,000,000 0.1150 0.2487 0.2690 0.1939 0.1048 0.0453 0.0163 0.0050 0.0014
400,000,000 0.1026 0.2337 0.2660 0.2018 0.1149 0.0523 0.0198 0.0065 0.0018
420,000,000 0.0916 0.2190 0.2617 0.2085 0.1246 0.0596 0.0237 0.0081 0.0024
440,000,000 0.0817 0.2047 0.2563 0.2139 0.1339 0.0671 0.0280 0.0100 0.0031
460,000,000 0.0730 0.1910 0.2500 0.2182 0.1428 0.0748 0.0326 0.0122 0.0040
480,000,000 0.0651 0.1779 0.2429 0.2212 0.1511 0.0825 0.0376 0.0147 0.0050
500,000,000 0.0581 0.1653 0.2352 0.2231 0.1587 0.0903 0.0428 0.0174 0.0062
520,000,000 0.0519 0.1534 0.2271 0.2240 0.1657 0.0981 0.0484 0.0205 0.0076
540,000,000 0.0463 0.1422 0.2185 0.2238 0.1720 0.1057 0.0541 0.0238 0.0091
560,000,000 0.0413 0.1316 0.2097 0.2228 0.1775 0.1131 0.0601 0.0274 0.0109
580,000,000 0.0369 0.1216 0.2008 0.2209 0.1823 0.1203 0.0662 0.0312 0.0129
600,000,000 0.0329 0.1123 0.1917 0.2182 0.1863 0.1272 0.0724 0.0353 0.0151
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 / 175,711,536 = 0.0000000057)
Pnotwin = (1.0 - Pwin) = 0.9999999943
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 )
For this example we will assume the cash value of the Jackpot is $120,000,000 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 .5660 0 0
1 .3221 120,000,000 38,652,000
2 .0917 60,000,000 5,502,000
3 .0174 40,000,000 696,000
4 .0025 30,000,000 75,000
5 .0003 24,000,000 7,200
Total 44,932,200
This Total then has to be divided by 1 - .5660 = .4340 to give a weighted Jackpot amount of 44,932,200 / .4340 = $103,530,415 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 0.7379 = $221,370,000 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 300,000,000 0.6274
20,000,000 0.9717 320,000,000 0.6072
40,000,000 0.9438 340,000,000 0.5875
60,000,000 0.9164 360,000,000 0.5685
80,000,000 0.8894 380,000,000 0.5501
100,000,000 0.8628 400,000,000 0.5323
120,000,000 0.8368 420,000,000 0.5150
140,000,000 0.8112 440,000,000 0.4984
160,000,000 0.7862 460,000,000 0.4824
180,000,000 0.7618 480,000,000 0.4669
200,000,000 0.7379 500,000,000 0.4520
220,000,000 0.7146 520,000,000 0.4376
240,000,000 0.6919 540,000,000 0.4238
260,000,000 0.6698 560,000,000 0.4105
280,000,000 0.6483 580,000,000 0.3977
300,000,000 0.6274 600,000,000 0.3854
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. A $56,000,000 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 $56,000,000 5.69115E-09 $0.3187
5 White No Mega 250,000 2.56102E-07 0.0640
4 White + Mega 10,000 1.45124E-06 0.0145
4 White No Mega 150 6.53059E-05 0.0098
3 White + Mega 150 7.25621E-05 0.0109
3 White No Mega 7 0.003265295 0.0229
2 White + Mega 10 0.001185181 0.0119
1 White + Mega 3 0.007111087 0.0213
0 White + Mega 2 0.013368843 0.0267
Total 0.025069987 0.5007
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.
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. The following analysis is based on the drawing for March 30, 2012 – the largest lottery Jackpot in world history.
We will use the Mega Millions drawing on March 30, 2012.
Quoted Jackpot: $656 million (Sum of annuity payments over 30 years)
Cash Jackpot: $473.6 million (Take the money now – used/valid for calculations)
Cash Jackpot after 40% discount for taxes: $284.16 million (Used for calculations)
Number of tickets sold: 651,915,940 (See http://www.lottoreport.com/mmsales.htm)
For the 3/30/2012 drawing there were 651,915,940 tickets sold. This number was plugged into the same Excel spreadsheet that was used to calculate the “Mult. Jackpot by this ratio for possible sharing” index. (See a couple of paragraphs back up.) The resulting index for 651,915,940 tickets is a tad under 0.3557. (Thank goodness for Excel.) This number is very close to the actual 3-way split that was the result of the drawing.
Finally, all prizes of $10,000 and above are reduced 40% to allow for federal and state taxes. Don’t forget that a large prize will throw you into a top tax bracket.
First, let’s calculate the effective cash Jackpot payout based on the nearly 652 million tickets in play. Thus: 0.3557- x 473,600,000 = 168,456,606. This is the before taxes, effective cash Jackpot amount, adjusted for the probability that you will have to share the Jackpot if you win. Then subtract 40% for taxes which will leave an after tax Jackpot of $101,073,964. Then multiply by the probability that you will win this Jackpot which yields: 101,073,964 x 5.691146E-09 = $0.5752 expected after tax return from the Jackpot.
Earlier we calculated a before tax expected return of $0.0640 for the regular “Match 5”. If we subtract 40% for taxes we get an after tax expected return of $0.0384. Similarly we previously found a before tax return of $0.0145 for “4 White + Mega”. Subtracting 40% for taxes leaves an after tax expected return of $0.0087. For all smaller prizes we assume that you don’t report your winnings. Thus we just add in the (0.0098 + 0.0109 + 0.0229 + 0.0119 + 0.0213 + 0.0267) = 0.1035.
Finally, to get the expected after tax return on your $1.00 ticket purchase, we just find the sum of all the above partial results. $0.5752 + 0.0384 + 0.0087 + 0.1035 = $0.7258. Thus, even for a huge Jackpot similar to the quoted $640 million for 3/30/2012, your after tax expected return is less than $0.73 for every $1.00 ticket that you buy.
The average return per $ 1.00 ticket includes the extremely low probability that you might win a large prize – for example $10,000 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:
50.05 % chance that you will get back $85 or less
59.64 % chance that you will get back $91 or less
70.08 % chance that you will get back $99 or less
80.02 % chance that you will get back $111 or less
89.90 % chance that you will get back $218 or less
95.01 % chance that you will get back $240 or less
98.01 % chance that you will get back $261 or less
99.00 % chance that you will get back $351 or less
99.50 % chance that you will get back $389 or less
99.80 % chance that you will get back $520 or less
Even if you buy 1,000 tickets, your chance of winning a $10,000 or larger prize is less than 0.2 %.
Government statistics (http://hazmat.dot.gov/riskmgmt/riskcompare.htm) show there are about 1.7 automobile caused fatalities for every 100,000,000 vehicle-miles. 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 1.7 / 100,000,000 = 0.000000034. This can also be stated as “One in 29,411,765-”. Thus, if you drive to the store to buy your Mega Millions ticket, your chance of being killed (or killing someone else) is nearly 6 times greater than the chance that you will win the Mega Millions Jackpot.
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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The numbers picked for the prizes consist of 5 numbers picked at random from a pool of 56 numbers (the White Numbers). Then a single number (the Mega Number) is picked from a second pool that has 46 numbers. If the results of these random number selections match one of the winning combinations on your lottery ticket, then you win something.
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 56 is: COMBIN(56,5) = 3,819,816. (See the math notation page or Help in Microsoft's Excel for more information on “COMBIN”).
For each of these 3,819,816 combinations there are COMBIN(46,1) = 46 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 3,819,816 x 46 = 175,711,536. 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: 1/175,711,536 = 0.00000000569114597006. If you express this as “One chance in ???”, you just divide “1” by the 0.00000000569114597006, which yields “One chance in 175,711,536”.
Match all 5 White numbers but not the Mega number (Payout = $250,000)
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 45 losing Mega numbers is: COMBIN(45,1) = 45. (Pick any of the 45 losers.) Thus there are COMBIN(5,5) x COMBIN(45,1) = 45 possible combinations. The probability for winning $250,000 is thus 45/175,711,536 = 0.000000256101568653 or “One chance in 3,904,700.80”.
Match 4 out of 5 White numbers and match the Mega number (Payout = $10,000)
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 51 losing White numbers is COMBIN(51,1) = 51. 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 this configuration: COMBIN(5,4) x COMBIN(51,1) x COMBIN(1,1) = 255. The probability of success is thus: 255/175,711,536 = 0.00000145124222237 or “One chance in 689,064.85”.
Match 4 out of 5 White numbers but not match the Mega number (Payout = $150)
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 51 losing White numbers is COMBIN(51,1) = 51. The number of ways your final number can match any of the 45 losing Mega numbers is: COMBIN(45,1) = 45. The product of these is the number of ways you can win this configuration: COMBIN(5,4) x COMBIN(51,1) x COMBIN(45,1) = 11,475. The probability of success is thus: 11,475/175,711,536 = 0.0000653059000065 or “One chance in 15,312.55”.
Match 3 out of 5 White numbers and match the Mega number (Payout = $150)
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 51 losing White numbers is COMBIN(51,2) = 1,275. 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 this configuration: COMBIN(5,3) x COMBIN(51,2) x COMBIN(1,1) = 12,750. The probability of success is thus: 12,750/175,711,536 = 0.0000725621111183 or “One chance in 13,781.30”.
Match 3 out of 5 White numbers but not match the Mega number (Payout = $7)
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 51 losing White numbers is COMBIN(51,2) = 1,275. The number of ways your final number can match any of the 45 losing Mega numbers is: COMBIN(45,1) = 45. The product of these is the number of ways you can win this configuration: COMBIN(5,3) x COMBIN(51,2) x COMBIN(45,1) = 573,750. The probability of success is thus: 573,750/175,711,536 = 0.00326529500032 or “One chance in 306.25”.
Match 2 out of 5 White numbers and match the Mega number (Payout = $10)
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 51 losing White numbers is COMBIN(51,3) = 20,825. 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 this configuration: COMBIN(5,2) x COMBIN(51,3) x COMBIN(1,1) = 208,250. The probability of success is thus: 208,250/175,711,536 = 0.00118518114827 or “One chance in 843.75”.
Match 1 out of 5 White numbers and match the Mega number (Payout = $3)
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 51 losing White numbers is COMBIN(51,4) = 249,900. 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 this configuration: COMBIN(5,1) x COMBIN(51,4) x COMBIN(1,1) = 1,249,500. The probability of success is thus: 1,249,500/175,711,536 = 0.00711108688959 or “One chance in 140.63”.
Match 0 out of 5 White numbers and match the Mega number (Payout = $2)
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 51 losing White numbers is COMBIN(51,5) = 2,349,060. 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 this configuration: COMBIN(5,0) x COMBIN(51,5) x COMBIN(1,1) = 2,349,060. The probability of success is thus: 2,349,060/175,711,536 = 0.0133688433524 or “One chance in 74.80”.
Probability of winning something
If we add all the ways you can win something we get:
1 + 45 + 255 + 11,475 + 12,750 + 573,750 + 208,250 + 1,249,500 + 2,349,060 = 4,405,086 different ways of winning something. If we divide by the 175,711,536, we get 0.0250699874367 as a probability of winning something. 1 divided by 0.0250699874367 yields “One chance in 39.89” 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 39.89. Thus, if the lottery officials proclaim that a given lottery drawing had 2 million “winners”, then there were about 2,000,000 x 39.89 ~= 79,780,000 tickets purchased that did not win the Jackpot. Alternately, there were about 79,780,000 - 2,000,000 ~= 77,780,000 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 1-56 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.)
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 0.0917 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 7 8
-------------------------------------------------------------------------------------
20,000,000 0.8924 0.1016 0.0058 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000
40,000,000 0.7964 0.1813 0.0206 0.0016 0.0001 0.0000 0.0000 0.0000 0.0000
60,000,000 0.7107 0.2427 0.0414 0.0047 0.0004 0.0000 0.0000 0.0000 0.0000
80,000,000 0.6343 0.2888 0.0657 0.0100 0.0011 0.0001 0.0000 0.0000 0.0000
100,000,000 0.5660 0.3221 0.0917 0.0174 0.0025 0.0003 0.0000 0.0000 0.0000
120,000,000 0.5051 0.3450 0.1178 0.0268 0.0046 0.0006 0.0001 0.0000 0.0000
140,000,000 0.4508 0.3592 0.1431 0.0380 0.0076 0.0012 0.0002 0.0000 0.0000
160,000,000 0.4023 0.3663 0.1668 0.0506 0.0115 0.0021 0.0003 0.0000 0.0000
180,000,000 0.3590 0.3678 0.1884 0.0643 0.0165 0.0034 0.0006 0.0001 0.0000
200,000,000 0.3204 0.3647 0.2075 0.0787 0.0224 0.0051 0.0010 0.0002 0.0000
220,000,000 0.2859 0.3580 0.2241 0.0935 0.0293 0.0073 0.0015 0.0003 0.0000
240,000,000 0.2552 0.3485 0.2380 0.1084 0.0370 0.0101 0.0023 0.0004 0.0001
260,000,000 0.2277 0.3369 0.2493 0.1230 0.0455 0.0135 0.0033 0.0007 0.0001
280,000,000 0.2032 0.3238 0.2580 0.1370 0.0546 0.0174 0.0046 0.0011 0.0002
300,000,000 0.1813 0.3096 0.2643 0.1504 0.0642 0.0219 0.0062 0.0015 0.0003
320,000,000 0.1618 0.2947 0.2684 0.1629 0.0742 0.0270 0.0082 0.0021 0.0005
340,000,000 0.1444 0.2795 0.2704 0.1744 0.0844 0.0326 0.0105 0.0029 0.0007
360,000,000 0.1289 0.2641 0.2705 0.1847 0.0946 0.0388 0.0132 0.0039 0.0010
380,000,000 0.1150 0.2487 0.2690 0.1939 0.1048 0.0453 0.0163 0.0050 0.0014
400,000,000 0.1026 0.2337 0.2660 0.2018 0.1149 0.0523 0.0198 0.0065 0.0018
420,000,000 0.0916 0.2190 0.2617 0.2085 0.1246 0.0596 0.0237 0.0081 0.0024
440,000,000 0.0817 0.2047 0.2563 0.2139 0.1339 0.0671 0.0280 0.0100 0.0031
460,000,000 0.0730 0.1910 0.2500 0.2182 0.1428 0.0748 0.0326 0.0122 0.0040
480,000,000 0.0651 0.1779 0.2429 0.2212 0.1511 0.0825 0.0376 0.0147 0.0050
500,000,000 0.0581 0.1653 0.2352 0.2231 0.1587 0.0903 0.0428 0.0174 0.0062
520,000,000 0.0519 0.1534 0.2271 0.2240 0.1657 0.0981 0.0484 0.0205 0.0076
540,000,000 0.0463 0.1422 0.2185 0.2238 0.1720 0.1057 0.0541 0.0238 0.0091
560,000,000 0.0413 0.1316 0.2097 0.2228 0.1775 0.1131 0.0601 0.0274 0.0109
580,000,000 0.0369 0.1216 0.2008 0.2209 0.1823 0.1203 0.0662 0.0312 0.0129
600,000,000 0.0329 0.1123 0.1917 0.2182 0.1863 0.1272 0.0724 0.0353 0.0151
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 / 175,711,536 = 0.0000000057)
Pnotwin = (1.0 - Pwin) = 0.9999999943
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
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 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 .5660 0 0
1 .3221 120,000,000 38,652,000
2 .0917 60,000,000 5,502,000
3 .0174 40,000,000 696,000
4 .0025 30,000,000 75,000
5 .0003 24,000,000 7,200
Total 44,932,200
This Total then has to be divided by 1 - .5660 = .4340 to give a weighted Jackpot amount of 44,932,200 / .4340 = $103,530,415 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 0.7379 = $221,370,000 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 300,000,000 0.6274
20,000,000 0.9717 320,000,000 0.6072
40,000,000 0.9438 340,000,000 0.5875
60,000,000 0.9164 360,000,000 0.5685
80,000,000 0.8894 380,000,000 0.5501
100,000,000 0.8628 400,000,000 0.5323
120,000,000 0.8368 420,000,000 0.5150
140,000,000 0.8112 440,000,000 0.4984
160,000,000 0.7862 460,000,000 0.4824
180,000,000 0.7618 480,000,000 0.4669
200,000,000 0.7379 500,000,000 0.4520
220,000,000 0.7146 520,000,000 0.4376
240,000,000 0.6919 540,000,000 0.4238
260,000,000 0.6698 560,000,000 0.4105
280,000,000 0.6483 580,000,000 0.3977
300,000,000 0.6274 600,000,000 0.3854
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. A $56,000,000 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 $56,000,000 5.69115E-09 $0.3187
5 White No Mega 250,000 2.56102E-07 0.0640
4 White + Mega 10,000 1.45124E-06 0.0145
4 White No Mega 150 6.53059E-05 0.0098
3 White + Mega 150 7.25621E-05 0.0109
3 White No Mega 7 0.003265295 0.0229
2 White + Mega 10 0.001185181 0.0119
1 White + Mega 3 0.007111087 0.0213
0 White + Mega 2 0.013368843 0.0267
Total 0.025069987 0.5007
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. The following analysis is based on the drawing for March 30, 2012 – the largest lottery Jackpot in world history.
We will use the Mega Millions drawing on March 30, 2012.
Quoted Jackpot: $656 million (Sum of annuity payments over 30 years)
Cash Jackpot: $473.6 million (Take the money now – used/valid for calculations)
Cash Jackpot after 40% discount for taxes: $284.16 million (Used for calculations)
Number of tickets sold: 651,915,940 (See http://www.lottoreport.com/mmsales.htm)
For the 3/30/2012 drawing there were 651,915,940 tickets sold. This number was plugged into the same Excel spreadsheet that was used to calculate the “Mult. Jackpot by this ratio for possible sharing” index. (See a couple of paragraphs back up.) The resulting index for 651,915,940 tickets is a tad under 0.3557. (Thank goodness for Excel.) This number is very close to the actual 3-way split that was the result of the drawing.
Finally, all prizes of $10,000 and above are reduced 40% to allow for federal and state taxes. Don’t forget that a large prize will throw you into a top tax bracket.
First, let’s calculate the effective cash Jackpot payout based on the nearly 652 million tickets in play. Thus: 0.3557- x 473,600,000 = 168,456,606. This is the before taxes, effective cash Jackpot amount, adjusted for the probability that you will have to share the Jackpot if you win. Then subtract 40% for taxes which will leave an after tax Jackpot of $101,073,964. Then multiply by the probability that you will win this Jackpot which yields: 101,073,964 x 5.691146E-09 = $0.5752 expected after tax return from the Jackpot.
Earlier we calculated a before tax expected return of $0.0640 for the regular “Match 5”. If we subtract 40% for taxes we get an after tax expected return of $0.0384. Similarly we previously found a before tax return of $0.0145 for “4 White + Mega”. Subtracting 40% for taxes leaves an after tax expected return of $0.0087. For all smaller prizes we assume that you don’t report your winnings. Thus we just add in the (0.0098 + 0.0109 + 0.0229 + 0.0119 + 0.0213 + 0.0267) = 0.1035.
Finally, to get the expected after tax return on your $1.00 ticket purchase, we just find the sum of all the above partial results. $0.5752 + 0.0384 + 0.0087 + 0.1035 = $0.7258. Thus, even for a huge Jackpot similar to the quoted $640 million for 3/30/2012, your after tax expected return is less than $0.73 for every $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 $10,000 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:
50.05 % chance that you will get back $85 or less
59.64 % chance that you will get back $91 or less
70.08 % chance that you will get back $99 or less
80.02 % chance that you will get back $111 or less
89.90 % chance that you will get back $218 or less
95.01 % chance that you will get back $240 or less
98.01 % chance that you will get back $261 or less
99.00 % chance that you will get back $351 or less
99.50 % chance that you will get back $389 or less
99.80 % chance that you will get back $520 or less
Even if you buy 1,000 tickets, your chance of winning a $10,000 or larger prize is less than 0.2 %.
2nd Thoughts
Government statistics (http://hazmat.dot.gov/riskmgmt/riskcompare.htm) show there are about 1.7 automobile caused fatalities for every 100,000,000 vehicle-miles. 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 1.7 / 100,000,000 = 0.000000034. This can also be stated as “One in 29,411,765-”. Thus, if you drive to the store to buy your Mega Millions ticket, your chance of being killed (or killing someone else) is nearly 6 times greater than the chance that you will win the Mega Millions Jackpot.
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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