

Updated for the game starting with the October 31, 2017 drawing. The new game picks 5 balls out of 70 and one ball out of 25.



Note:

See











Updated for the game starting with the October 31, 2017 drawing. The new game picks 5 balls out of 70 and one ball out of 25.Note:See http://www.durangobill.com/OldMegaMillionsOdds.html for the version of the game that was valid before October 31, 2017.

Concise Table of Mega Millions Odds

(Mathematical derivation below)







Ticket Matches Payout Odds Probability

---------------------------------------------------------------------

5 White + Mega Jackpot 1 in 302,575,350.00 0.000000003305

5 White No Mega 1,000,000 1 in 12,607,306.25 0.00000007932

4 White + Mega 10,000 1 in 931,001.08 0.000001074

4 White No Mega 500 1 in 38,791.71 0.00002578

3 White + Mega 200 1 in 14,546.89 0.00006874

3 White No Mega 10 1 in 606.12 0.001650

2 White + Mega 10 1 in 692.71 0.001444

1 White + Mega 4 1 in 89.38 0.01119

0 White + Mega 2 1 in 36.63 0.02730



Win something Variable 1 in 23.99 0.04168





Game Rules

In the new Oct. 2017 version, the price of a ticket has been increased to $2 from $1. The numbers picked for the prizes consist of 5 numbers picked at random from a pool of 70 numbers (the White Numbers). Then a single number (the Mega Number) is picked from a second pool that has 25 numbers. If the results of these random number selections match one of the winning combinations on your lottery ticket, then you win something.





Cash Value of a Game

If you win the Jackpot, you have the option of either accepting 30 annuity payments (A cash amount that is paid annually), or you can take a lump sum "cash value" payment. If you take the annuity, you will get a series of payments that will increase at 5% per year. The total of these 30 payments will approximately equal the advertised size of the Jackpot. (This assumes you are a single winner and not sharing the Jackpot with someone else. Taxes will subsequently be subtracted/withheld from these payments.)



If you take an immediate cash payment instead of the 30 year annuity, you will get about one half (before taxes) of the advertised "annuity" value of the game. Lower interest rates will increase this fraction while higher interest rates will decrease the cash payout.



The following table shows the relative amounts that you would receive given various interest rates. For the calculations below, an advertised "annuity" value for the Jackpot of $100,000,000 is assumed. (The $100,000,000 could be your share of a shared Jackpot.) The amounts in the table can be directly scaled for any arbitrary Jackpot size.



(All amounts are BEFORE taxes.)





Interest Immediate First Payment Last Payment

Rate Cash Payout If Take Annuity If Take Annuity



3.0% $60,506,809 $1,505,144 $6,195,375

3.5% 56,062,409 1,505,144 6,195,375

4.0% 52,053,422 1,505,144 6,195,375

4.5% 48,431,551 1,505,144 6,195,375

5.0% 45,154,305 1,505,144 6,195,375

6.0% 39,488,405 1,505,144 6,195,375

8.0% 30,912,469 1,505,144 6,195,375



(All results rounded to the nearest $1)



Note that the cash value of the Jackpot is normally about one half of the advertised value of the Jackpot. For the rest of this web page, we will just refer to these exact values as "about one half" instead of the above exact values.



As an example of scaling for different sized Jackpots, if the interest rate is 4% and the advertised annuity Jackpot is $300,000,000, then multiply the results in the 4% row by 3 to get the corresponding instant cash payout amount or annual annuity payout amounts.





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 70 is: COMBIN(70,5) = 12,103,014. (See the



For each of these (All results rounded to the nearest $1)Note that the cash value of the Jackpot is normally about one half of the advertised value of the Jackpot. For the rest of this web page, we will just refer to these exact values as "about one half" instead of the above exact values.As an example of scaling for different sized Jackpots, if the interest rate is 4% and the advertised annuity Jackpot is $300,000,000, then multiply the results in the 4% row by 3 to get the corresponding instant cash payout amount or annual annuity payout amounts.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.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 70 is: COMBIN(70,5) = 12,103,014. (See the math notation page or Help in Microsoft's Excel for more information on “COMBIN”).For each of these

12,103,014 combinations there are COMBIN(25,1) = 25 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 two partial calculations. Thus, the total number of equally likely Mega Millions combinations is 12,103,014 x 25 = 302,575,350. 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: 1/302,575,350 = 0.000000003304961888. If you express this as “One chance in ???”, you just divide “1” by the 0.000000003304961888, which yields “One chance in 302,575,350”.



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 24 losing Mega numbers is: COMBIN(24,1) = 24. (Pick any of the 24 losers.) Thus there are COMBIN(5,5) x COMBIN(24,1) = 24 possible combinations. The probability for winning $1,000,000 is thus 24/302,575,350 = .0000000793190853121 or “One chance in 12,607,306.25”.



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 65 losing White numbers is COMBIN(65,1) = 65. 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(65,1) x COMBIN(1,1) = 325. The probability of success is thus: 325/302,575,350 = 0.0000010741126136 or “One chance in 931,001.08”.



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 65 losing White numbers is COMBIN(65,1) = 65. The number of ways your final number can match any of the 24 losing Mega numbers is: COMBIN(24,1) = 24. The product of these is the number of ways you can get this configuration: COMBIN(5,4) x COMBIN(65,1) x COMBIN(24,1) = 7,800. The probability of success is thus: 7,800/302,575,350 = 0.0000257787 or “One chance in 38,791.71”.



Match 3 out of 5 White numbers and match the Mega number (Payout = $200)

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 65 losing White numbers is COMBIN(65,2) = 2,080. 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(65,2) x COMBIN(1,1) = 20,800. The probability of success is thus: 20,800/302,575,350 = 0.0000687432 or One chance in 14,546.89”.



Match 3 out of 5 White numbers but not match the Mega number (Payout = $10)

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 65 losing White numbers is COMBIN(65,2) = 2,080. The number of ways your final number can match any of the 24 losing Mega numbers is: COMBIN(24,1) = 24. The product of these is the number of ways you can get this configuration: COMBIN(5,3) x COMBIN(65,2) x COMBIN(24,1) = 499,200. The probability of success is thus: 499,200/

302,575,350 = 0.001649837 or “One chance in 606.12”.



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 65 losing White numbers is COMBIN(65,3) = 43,680. 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(65,3) x COMBIN(1,1) = 436,800. The probability of success is thus: 436,800/302,575,350 = 0.001443607 or “One chance in 692.71”.



Match 1 out of 5 White numbers and match the Mega number (Payout = $4)

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 65 losing White numbers is COMBIN(65,4) = 677,040. 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(65,4) x COMBIN(1,1) = 3,385,200. The probability of success is thus: 3,385,200/302,575,350 = 0.011187957 or “One chance in 89.38”.



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 65 losing White numbers is COMBIN(65,5) = 8,259,888. 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(65,5) x COMBIN(1,1) = 8,259,888. The probability of success is thus: 8,259,888/302,575,350 = 0.0272986 or “One chance in 36.63”.



Probability of winning something

If we add all the ways you can win something we get:

1 + 24 + 325 + 7,800 + 20,800 + 499,200 + 436,800 + 3,385,200 + 8,259,888 = 12,610,038 different ways of winning something. If we divide the result by the 302,575,350, we get .0416757 as a probability of winning something. 1 divided by 0.0416757 yields “One chance in 23.99” of winning something. Alternately, the probability of not winning anything is 1 - .0416757 = 0.9583243 which is about 23 out of every 24 tickets.







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 23.99. Thus, if the lottery officials proclaim that a given lottery drawing had 5 million “winners”, then there were about 5,000,000 x 23.99 ~= 120,000,000 tickets purchased that did not win the Jackpot. ("~=" means "approximately equal to") Alternately, there were about 120,000,000 - 5,000,000 ~= 115,000,000 tickets that did not win anything.





Probability of not matching anything

Match 0 out of 5 White numbers and not match the Mega number

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 65 losing White numbers is COMBIN(65,5) = 8,259,888. The number of ways your final number can fail to match the Mega number is: COMBIN(24,1) = 24. The product of these numbers is the number of ways you can get this configuration: COMBIN(5,0) x COMBIN(65,5) x COMBIN(24,1) = 198,237,312. The probability of failing to match anything is thus: 198,237,312/302,575,350 = 0.65516676 or just under two times out of every 3 tickets.



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 1-70 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 200,000,000 tickets are in play for a Mega Millions game, then there is a 0.1128 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, then find the difference in the annuity Jackpot amount for the prior game vs. the current annuity Jackpot amount. The number of tickets in play is approximately three quarters (0.75) of this number. (Technically, this "3/4" is variable. If interest rates are low, the ratio will be greater than 3/4. And vice versa, if interest rates are high, the ratio will be less than 3/4.)



For example, if the preceding game had an advertised annuity payout amount of $200,000,000 and the current game has an advertised annuity payout amount of $280,000,000, then there are about (280,000,000 - 200,000,000) x 0.75 = 80,000,000 x 0.75 = 60,000,000 tickets in play for the current game. (Past Jackpot amounts can be seen at:



The table below shows the probabilities for 0 to 6 jackpot winners for various numbers of tickets in play. The graph above is derived from a larger version of this table.



“N” Number “K”

of tickets Number of tickets holding the Jackpot combination

in play 0 1 2 3 4 5 6

----------------------------------------------------------------------

100,000,000 0.7186 0.2375 0.0392 0.0043 0.0004 0.0000 0.0000

200,000,000 0.5163 0.3413 0.1128 0.0249 0.0041 0.0005 0.0001

300,000,000 0.3710 0.3679 0.1824 0.0603 0.0149 0.0030 0.0005

400,000,000 0.2666 0.3524 0.2330 0.1027 0.0339 0.0090 0.0020

500,000,000 0.1916 0.3166 0.2616 0.1441 0.0595 0.0197 0.0054





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 / 302,575,350 = 0.000000003305)

Pnotwin = (1.0 - Pwin) = 0.999999996695

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 )



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 200,000,000 tickets are in play for a Mega Millions game, then there is a 0.1128 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, then find the difference in the annuity Jackpot amount for the prior game vs. the current annuity Jackpot amount. The number of tickets in play is approximately three quarters (0.75) of this number. (Technically, this "3/4" is variable. If interest rates are low, the ratio will be greater than 3/4. And vice versa, if interest rates are high, the ratio will be less than 3/4.)For example, if the preceding game had an advertised annuity payout amount of $200,000,000 and the current game has an advertised annuity payout amount of $280,000,000, then there are about (280,000,000 - 200,000,000) x 0.75 = 80,000,000 x 0.75 = 60,000,000 tickets in play for the current game. (Past Jackpot amounts can be seen at: http://www.lottoreport.com/ticketcomparison.htm The cash payout value for these lottoreport numbers would be about one-half the announced Jackpot annuity amount.)The table below shows the probabilities for 0 to 6 jackpot winners for various numbers of tickets in play. The graph above is derived from a larger version of this table.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 playK = Number of tickets holding the Jackpot combinationPwin = Probability that a random ticket will win ( = 1 / 302,575,350 = 0.000000003305)Pnotwin = (1.0 - Pwin) = 0.999999996695COMBIN(N,K) = number of ways to select K items from a group of N itemsx = multiply terms^ = raise to power (e.g. 2^3 = 8 )









Sample Calculation to Find the Expected Cash Ticket Value

Considering the Number of Tickets That are in Play





For this example we will assume the cash value of the Jackpot is $400,000,000 (The advertised value of Jackpot would be about double this.) and there are 300,000,000 tickets in play for the current game. Probability values are from the “300,000,000” row above. The calculated value of a ticket includes the possibility that you will share the Jackpot with someone else.



The first calculation is: “What is the probability that the jackpot will be won?” This is simply (1.00 – the probability that no one will win) = 1.00 – 0.3710 ~= 0.6290. Thus the expected cash Jackpot payout by the lottery is $400,000,000 times 0.6290 = $251,590,404.62. (All actual calculations used full precision)



If there are 300,000,000 tickets in play, then we divide the $251,590,404.62 by 300,000,000 to get an average cash Jackpot payout per ticket of $0.8386. The other smaller prizes add $0.24698 to this amount to give an "expected before tax, cash value of $1.0856. (And remember, you paid $2 for this ticket.)



These calculations can be used to form a table that shows the expected cash return per ticket ( = expected cash value per ticket). For example if the cash value of the jackpot is $400,000,000 and there are 300,000,000 tickets in play, then the ticket’s expected cash value is fractionally under $1.09.



The following table shows the "Expected Before Taxes Cash Value" of a $2.00 ticket. (includes $0.24698 for the smaller prizes)





Nbr. Tickets

In Play < - - - - Cash Jackpot Size in Millions - - - - >

In Millions 100 200 300 400 500 600 700 800 900 1000

-----------------------------------------------------------------------

100 0.53 0.81 1.09 1.37 1.65 1.94 2.22 2.50 2.78 3.06

200 0.49 0.73 0.97 1.21 1.46 1.70 1.94 2.18 2.42 2.67

300 0.46 0.67 0.88 1.09 1.30 1.50 1.71 1.92 2.13 2.34

400 0.43 0.61 0.80 0.98 1.16 1.35 1.53 1.71 1.90 2.08

500 0.41 0.57 0.73 0.89 1.06 1.22 1.38 1.54 1.70 1.86

600 0.39 0.53 0.68 0.82 0.97 1.11 1.25 1.40 1.54 1.68

700 0.38 0.50 0.63 0.76 0.89 1.02 1.15 1.28 1.41 1.53

800 0.36 0.48 0.60 0.71 0.83 0.94 1.06 1.18 1.29 1.41

900 0.35 0.46 0.56 0.67 0.77 0.88 0.99 1.09 1.20 1.30

1000 0.34 0.44 0.54 0.63 0.73 0.82 0.92 1.02 1.11 1.21



The calculations above use the "cash value" of the Jackpot. The "cash value" is usually about one half of the advertised Jackpot amount. Thus the "expected value" of your $2 ticket will be on the left side of the table unless the advertised Jackpot is over One $Billion. (That's a "B")



We can also see what happens to the expected value of a ticket if a buying frenzy should develop at this point. Let’s assume that 400 million more tickets are sold. At $2,00 per ticket, the lottery takes in $800 million. This $800 million is divvied up multiple ways. (For example, see the 6 "winners" in the "3rd Thoughts" section at the end of this web page, allocations for the smaller prizes, etc.)



Only about $200 million of the new cash input goes to the Jackpot allocation. (The rate of increase in Jackpot amount for a given new cash input is based on historical Mega Million data which can be seen at



Thus the game has been transformed into 700 million tickets in play for a cash jackpot that is now worth $600 million. If we follow the 700 million ticket row to the right until we reach the $600 million column, we find an expected cash jackpot value of $1.02. The buying frenzy has decreased the expected cash value of a ticket from $1.09 to $1.02. And remember, you paid $2.00 cash for the "privilege" of participating in this (con) game.











For this example we will assume the cash value of the Jackpot is $400,000,000 (The advertised value of Jackpot would be about double this.) and there are 300,000,000 tickets in play for the current game. Probability values are from the “300,000,000” row above. The calculated value of a ticket includes the possibility that you will share the Jackpot with someone else.The first calculation is: “What is the probability that the jackpot will be won?” This is simply (1.00 – the probability that no one will win) = 1.00 – 0.3710 ~= 0.6290. Thus the expected cash Jackpot payout by the lottery is $400,000,000 times 0.6290 = $251,590,404.62. (All actual calculations used full precision)If there are 300,000,000 tickets in play, then we divide the $251,590,404.62 by 300,000,000 to get an average cash Jackpot payout per ticket of $0.8386. The other smaller prizes add $0.24698 to this amount to give an "expected before tax, cash value of $1.0856. (And remember, you paid $2 for this ticket.)These calculations can be used to form a table that shows the expected cash return per ticket ( = expected cash value per ticket). For example if the cash value of the jackpot is $400,000,000 and there are 300,000,000 tickets in play, then the ticket’s expected cash value is fractionally under $1.09.The following table shows the "Expected Before Taxes Cash Value" of a $2.00 ticket. (includes $0.24698 for the smaller prizes)The calculations above use the "cash value" of the Jackpot. The "cash value" is usually about one half of the advertised Jackpot amount. Thus the "expected value" of your $2 ticket will be on the left side of the table unless the advertised Jackpot is over One $Billion. (That's a "B")We can also see what happens to the expected value of a ticket if a buying frenzy should develop at this point. Let’s assume that 400 million more tickets are sold. At $2,00 per ticket, the lottery takes in $800 million. This $800 million is divvied up multiple ways. (For example, see the 6 "winners" in the "3rd Thoughts" section at the end of this web page, allocations for the smaller prizes, etc.)Only about $200 million of the new cash input goes to the Jackpot allocation. (The rate of increase in Jackpot amount for a given new cash input is based on historical Mega Million data which can be seen at http://www.lottoreport.com/mmsales.htm ) (The increase in the advertised annuity would be about double this - or about $400 million.) The cash Jackpot is now worth $400 million plus $200 million = $600 million. (The advertised annuity value would be about double this - or about 1.2 Billion.)Thus the game has been transformed into 700 million tickets in play for a cash jackpot that is now worth $600 million. If we follow the 700 million ticket row to the right until we reach the $600 million column, we find an expected cash jackpot value of $1.02. The buying frenzy has decreased the expected cash value of a ticket from $1.09 to $1.02. And remember, you paid $2.00 cash for the "privilege" of participating in this (con) game.

Before Tax Return on Investment





It is interesting to calculate what the long term expected return is for each $2.00 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 average $228,000,000 cash payout (decline the annuity) is assumed for the Jackpot. (Could be your portion of a shared Jackpot.) (Note that this "cash payout" is about one half of the advertised Jackpot amount.



Combination Cash Payout Probability Contribution

-----------------------------------------------------------------

5 White + Mega $228,000,000 3.30496E-09 $0.7535

5 White No Mega 1,000,000 7.93191E-08 0.0793

4 White + Mega 10,000 1.07411E-06 0.0107

4 White No Mega 500 2.57787E-05 0.0129

3 White + Mega 200 6.87432E-05 0.0137

3 White No Mega 10 0.001649837 0.0165

2 White + Mega 10 0.001443607 0.0144

1 White + Mega 4 0.011187957 0.0448

0 White + Mega 2 0.027298615 0.0546



Total 0.025069987 1.0005



Total for last 6 rows 0.1569

(Used for after tax calculation)



Total for last 8 rows 0.24698

(Used for before taxes, non Jackpot, cash value calculations)



Thus, for each $2.00 that you spend for Mega Millions tickets, you can expect to get back about $1.00. Of course you get to pay taxes on any large payout, so your net return is even less.













Expected after tax return on your $2.00 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 huge Jackpot exists. The following analysis assumes the annuity value of the Jackpot is $2 Billion (that’s a “B”) and there are 600 million tickets in play. The cash value for any Jackpot is about one-half the annuity value which brings the real value down to $1,000,000,000. 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, we check the expected value of a ticket in the table that we calculated earlier. Follow the 600-million row until you come to the $1,000 million column. The expected cash value of a ticket is $1.68. This included $0.1569 for the smaller prizes (no taxes for prizes under $10,000) so $0.1569 has to be subtracted back out. This leaves $1.53 for the 3 largest prizes. However, this has to be reduced by 40% for taxes. This leaves an expected after tax value of the jackpot of $0.9164. Finally we add the $0.1569 back in again to get an expected after tax return of $1.07 for your $2 cost of your ticket. And this is for an advertised $2 Billion Jackpot.



The following shows the actual numbers for the Oct. 23, 2018 Mega Millions game.

Announced annuity value: $1.6 billion

Actual cash value: $903 million (Interest rates were still relatively low)

Number of tickets in play: 370 million (Less than the usual buying frenzy)



The relatively higher cash value and relatively low number of tickets should have "enhanced the expected value".



The actual calculated expected after tax value was still only $1.24 for a $2.00 ticket. For this particular game, the expected after tax loss for every $2 ticket that you bought was $0.76.







Megaplier





Some states use a Megaplier feature to increase non-jackpot prizes by 2, 3, 4 or 5 times. It costs an additional $1 for the Megaplier play. The Megaplier number is randomly picked using the probability table below.



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 3 0.333333 0.666667

3 2 in 5 (1 in 2.5) 0.4 1.2

4 1 in 5 0.2 0.8

5 1 in 15 0.0666667 0.333333

Totals 1.000000 3.0



Contribution = “Multiple” times “Probability”



1.0 has to be subtracted from this 3.0 because you would win “1 unit” of the sub prizes just from your simple ticket purchase. This leaves a “bonus contribution multiplier” of just 2.0.



Thus we have calculated that the average multiplier is 2.0. We then multiply the average extra expected return for all the sub-prizes (previously calculated) by this 2.0 to get the expected return if you buy the “Megaplier” option.



(0.0793 + 0.0107 + 0.0129 + 0.0137 + 0.0165 + 0.0144 + 0.0448 + 0.0546 = 0.24698) x 2.0 = $0.494



Thus if you pay another $1.00 to buy the Megaplier option, your expected before tax return is $0.494. (The after tax return would be less than this.)









Just the Jackpot





The new version of Mega Millions provides still another way to play the game.

You can pay $3 for 2 quick picks that only participate in the Jackpot prize. (You don't have to worry about matching any of the other combinations.)



If there is an advertised Jackpot of $1 billion then the cash value of the Jackpot is about $500 million. If there are 500,000 tickets in play, the expected before tax value of one ticket for just the Jackpot only is 0.808426. However you get 2 tickets which brings the expected value up to $1.616852. The bad news is that you paid $3 to get something worth $1.62. If you win, you will pay 40% of your winnings back in taxes - which leaves an expected after tax value of $0.97. (And you paid $3 for this privilege.)













Percentile Expected Returns on Ticket Purchases





The average return per $2.00 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 500 tickets ( = 1 ticket for each of 500 Mega Millions games = 1 ticket per Mega Millions game, 2 times a week for 4.8 years) there is a:



49.79 % chance that you will get back $64 or less

58.41 % chance that you will get back $68 or less

69.82 % chance that you will get back $74 or less

78.81 % chance that you will get back $80 or less

89.56 % chance that you will get back $92 or less

94.96 % chance that you will get back $116 or less

98.04 % chance that you will get back $280 or less

99.02 % chance that you will get back $554 or less

99.52 % chance that you will get back $572 or less

99.90 % chance that you will get back $750 or less



Even if you buy 500 tickets, your chance of winning a $10,000 or larger prize is less than 1 chance in 1,000.

















2nd Thoughts

The 2015 fatality rate per 100 million Vehicle Miles Traveled in the U.S. was 1.12.



Alternately, if you “played” Russian Roulette 100 times per day, every day for 83 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)



The 2015 fatality rate per 100 million Vehicle Miles Traveled in the U.S. was 1.12. https://crashstats.nhtsa.dot.gov/Api/Public/ViewPublication/812318 . 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 1.12 / 100,000,000 = 0.0000000224 fatalities per vehicle mile. This can also be stated as “One chance in 44,642,857”. Thus, if you drive one mile to (and return from) the store to buy your Mega Million ticket, your chance of being killed (or killing someone else) is nearly 7 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 83 years, with Mega Millions Jackpot odds, you would have better than a 99% chance of surviving.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 Government2) State Government (Taxes)3) State Government (Direct share)4) Merchants that sell tickets5) Lottery companies6) Advertisers and promotersAnd the losers are:(Mathematically challenged and proud of it)

Also please see the related calculations for

Powerball

.















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