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 Powerball 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 Powerball game, then there is a 0.1181 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 Powerball 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. A history of these past jackpot amounts (subtract about 50 % from the stated jackpot amount to get the cash payout) can be seen at: http://www.lottostrategies.com/script/jackpot_history/draw_date/101)

“N”   Number                           “K”
of tickets        Number of tickets holding the Jackpot combination
in play            0       1       2       3       4       5       6
----------------------------------------------------------------------
20,000,000      0.8721  0.1194  0.0082  0.0004  0.0000  0.0000  0.0000
40,000,000      0.7605  0.2082  0.0285  0.0026  0.0002  0.0000  0.0000
60,000,000      0.6632  0.2724  0.0559  0.0077  0.0008  0.0001  0.0000
80,000,000      0.5784  0.3167  0.0867  0.0158  0.0022  0.0002  0.0000
100,000,000     0.5044  0.3452  0.1181  0.0270  0.0046  0.0006  0.0001
120,000,000     0.4399  0.3613  0.1484  0.0406  0.0083  0.0014  0.0002
140,000,000     0.3836  0.3675  0.1761  0.0562  0.0135  0.0026  0.0004
160,000,000     0.3345  0.3663  0.2006  0.0732  0.0200  0.0044  0.0008
180,000,000     0.2917  0.3594  0.2214  0.0909  0.0280  0.0069  0.0014
200,000,000     0.2544  0.3482  0.2383  0.1088  0.0372  0.0102  0.0023
220,000,000     0.2219  0.3341  0.2515  0.1262  0.0475  0.0143  0.0036
240,000,000     0.1935  0.3178  0.2610  0.1429  0.0587  0.0193  0.0053
260,000,000     0.1687  0.3002  0.2671  0.1585  0.0705  0.0251  0.0074
280,000,000     0.1471  0.2820  0.2702  0.1726  0.0827  0.0317  0.0101
300,000,000     0.1283  0.2635  0.2705  0.1851  0.0950  0.0390  0.0134
320,000,000     0.1119  0.2451  0.2684  0.1959  0.1073  0.0470  0.0172
340,000,000     0.0976  0.2271  0.2642  0.2049  0.1192  0.0555  0.0215
360,000,000     0.0851  0.2097  0.2583  0.2122  0.1307  0.0644  0.0264
380,000,000     0.0742  0.1930  0.2510  0.2176  0.1415  0.0736  0.0319
400,000,000     0.0647  0.1772  0.2425  0.2213  0.1515  0.0829  0.0378

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 / 146,107,962  =  0.0000000068)
Pnotwin = (1.0 - Pwin)  = 0.9999999932
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 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              .5044                     0                  0
1              .3452           120,000,000         41,424,000
2              .1181            60,000,000          7,086,000
3              .0270            40,000,000          1,080,000
4              .0046            30,000,000            138,000
5              .0006            24,000,000             14,400
6              .0001            20,000,000              2,000
Total                                              49,744,400

This total then has to be divided by 1 - .5044 = .4956 to give a weighted Jackpot amount of  49,744,400 / .4956 = $100,372,074 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  0.6912 = $207,360,000 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          0.6912
 20,000,000            0.9660          220,000,000          0.6647
 40,000,000            0.9327          240,000,000          0.6390
 60,000,000            0.8999          260,000,000          0.6141
 80,000,000            0.8678          280,000,000          0.5901
100,000,000            0.8364          300,000,000          0.5669
120,000,000            0.8058          320,000,000          0.5446
140,000,000            0.7759          340,000,000          0.5230
160,000,000            0.7469          360,000,000          0.5023
180,000,000            0.7186          380,000,000          0.4822
200,000,000            0.6912          400,000,000          0.4630




   The Powerball game includes an optional “Multiplier”. If you spend an extra $1 for the multiplier, then any payout except the jackpot is multiplied by whatever number shows up when the televised “Multiplier Wheel” is spun. The “Multiplier Wheel” has four 2’s, four 3’s, four 4’s, and four 5’s. The net effect of the “multiplier” is found by multiplying the probability of each outcome by the resulting digit, adding the results together, and then subtracting 1.00. (1.00 is subtracted as you would get this payout even if you just played the regular game.) Thus we can calculate the weighted multiplier amount as follows:
Weighted Multiplier = 0.25 x 2 + 0.25 x 3 + 0.25 x 4 + 0.25 x 5 – 1.00 = 2.5
We will use this result in the “Return on Investment” section.

   Occasionally, a “gimmick” is introduced to the Power Play Multiplier where one of the 5’s is replaced by a 10. When this happens, the Weighted Multiplier becomes:
Weighted Multiplier = 0.25 x 2 + 0.25 x 3 + 0.25 x 4 + 0.1875 x 5 + 0.0625 x 10 - 1.00 = 2.8125. If you want to calculate the “Return on Investment” when this gimmick is in play, use the above 2.8125 instead of 2.5.
   Finally, it is interesting to calculate what the long term expected return is for each $1.00 lottery ticket that you buy. We will also calculate the return on the optional Power Play multiplier.

   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 $44,000,000 cash payout (decline the annuity) is assumed for the Jackpot. (Could be your portion of a shared Jackpot.)

Combination       Payout     Probability   Contribution
-------------------------------------------------------
5 White + PB   $44,000,000   6.84425E-09     $0.3011
5 White No PB      200,000   2.80614E-07      0.0561
4 White + PB        10,000   1.71106E-06      0.0171
4 White No PB          100   7.01536E-05      0.0070
3 White + PB           100   8.38421E-05      0.0084
3 White No PB            7   0.003437527      0.0241
2 White + PB             7   0.001341474      0.0094
1 White + PB             4   0.007881158      0.0315
PB                       3   0.014501332      0.0435

Total                        0.027317485      0.4983
Total for last 8 rows                         0.1971

   Thus, for each $1.00 that you spend for Powerball 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.

   Next, we can calculate the expected return if you pay another $1.00 for the “Power Play Multiplier”. Here we use the $0.1971 from the last 8 rows as the multiplier is used for all payouts except the Jackpot. When we multiply the $0.1971 by the “Weighted Multiplier” of 2.5 that we calculated earlier, we get: 0.1971 x 2.5 = $0.4928. Thus, for each $1.00 that you pay for the “Power Play Multiplier”, your long run expected return is to get back about 49.3 cents.

   While the above calculation represents an average Powerball 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 10/19/05. The advertised Jackpot was $340 million (annuity total). According to the official MUSL commission, the cash value (net present value before taxes) was $164,410,058. Also there was a carryover bonus “Match 5” pool of $653,492 x 49 = $32,021,108 that would be split between ticket holders that matched the 5 white balls but did not match the Powerball. This “Match 5 Bonus” pool is paid if and only if someone wins the Jackpot.

   For the 10/19/05 drawing there were over 160 million tickets in play. For the following calculations, this is rounded down to 160 million. (The actual larger number of tickets in play would slightly reduce the calculated return on investment shown below.) 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 Jackpot payout based on 160 million tickets in play. (Please see the “Shared Jackpot Amount When a Large Number of Tickets are in Play” section for the calculation method, but we will use the 160 million row.) Thus:
(0.3663 x 164410058 + 0.2006 x 164410058/2 + 0.0732 x 164410058/3 + 0.0200 x 164410058/4 + 0.0044 x 164410058/5 + 0.0008 x 164410058/6) / (1 - 0.3345) = $122,785,862. This is the before taxes, effective cash Jackpot amount, adjusted for the possibility that you will have to share the Jackpot if you win. Then subtract 40% for taxes which will leave an after tax Jackpot of $73,671,517. Then multiply by the probability that you will win this Jackpot which yields: 73671517 x 6.84425E-09 = $0.5042 expected after tax return from the Jackpot.

   Next, we calculate the expected after tax return from the “Match 5 Bonus”. With 160 million tickets in play, the expected number of tickets that would match the 5 white balls but not match the Powerball is 160000000 x 41/146107962 (calculated earlier) ~= 45 tickets that would share the “Match 5 Bonus”. (In actuality, there were 49 of these) Thus the expected bonus would be: $32,021,108 / 45 = $711,580. (The actual number of “Match 5” winners was larger - thus reducing the size of each share.) This 711,580 is paid if and only if someone wins the Jackpot. Since there would be a 0.3345 probability that no one would win the Jackpot, this is reduced by 33.45% to $473,556. Then when we subtract another 40% for taxes we get $284,134. There is a 41/146107962 probability that your ticket will match the 5 winning numbers which finally yields a 284,134 x 41 / 146107962 = 0.0797 expected after tax return from the “Match 5 Bonus”.

   Earlier we calculated a before tax expected return of $0.0561 for the regular “Match 5”. If we subtract 40% for taxes we get an after tax expected return of $0.0337. Similarly we previously found a before tax return of $0.0171 for “4 White + PB”. Subtracting 40% for taxes leaves an after tax expected return of $0.0103. For all smaller prizes we assume that you don’t report your winnings. Thus we just add in the (0.0070 + 0.0084 + 0.0241 + 0.0094 + 0.0315 + 0.0435) = 0.1239

   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. $0.5042 +  0.0797 + 0.0337 + 0.0103 + 0.1239 = $0.7518. Thus, even for a huge Jackpot similar to the quoted $340 million for 10/19/05, your after tax expected return is only about $0.75 for every $1.00 ticket that you buy.


   Government statistics 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 lottery 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 Powerball ticket, your chance of being killed (or killing someone else) is about 5 times greater than the chance that you will win the Powerball 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 Mega Millions



   For reasons unknown and for which Yahoo refuses to disclose, this entire website has been blacklisted/banned by Yahoo’s search engine. Other websites have suffered a similar fate. If you are trying to find information via Google’s search engine vs. Yahoo’s search engine, you should understand that Yahoo’s results may not include the information that you are seeking.
Return to Durango Bill's Home page