When do Powerball or Mega Millions actually return you money?
With the Powerball Lottery having recently reached $1.58 billion, there are lots of useful posts about calculating the expected value of your money. Everyone already knows about how many bankrupt lotto winners there are. Some people have written about about buying every ticket. And others have written about how the annuity is a better value than you might think.
Calculating the return on the lottery is mostly straight forward and there are many (so many, oh so very many) places that calculate this. The one tricky part is how to estimate the number of tickets sold. Some articles only mention this and others try to estimate it. I present a new methodology that can easily estimate this, hopefully in a better way.
What is Expected Value
Expected Value is the average result of something if it is repeated an infinite number of times. Since infinite is hard to do, it can be calculated by taking an average of each distinct result multiplied by how probable it is. In context of the lottery, we are taking the average of each type of payout multiplied by its probability. And after calculating the expected value, this can be compared to the price of the ticket. If the expected value is more than the price of the ticket, you are expected to win more money than you spend. All casino games (with the exception of possibly table poker) have a lower expected value than the cost of playing. All lottery games do also, even this $1.58 billion Powerball lottery as I will explain below.
Odds of winning
The odds of winning Powerball and Mega Millions are well known. The odds of hitting the jackpot is easy to calculate. We take the number of combinations of the 5 numbers multiplied by the number of Powerball / Megaball numbers. Specifically
Unfortunately, calculating the odds of other prizes are more complicated. There is a prize for only hitting the special ball (Powerball or Megaball). So you might think the odds of this would be 1 in 26/15, the range of the Powerball/Megaball number. But this is incorrect because this also includes the possibility of hitting one of the other prizes (special ball and some additional numbers). So these other winning combinations need to be removed to find the odds of hitting the special ball alone. Calculating these can be tricky and is not the point of this article..
Strangely enough, both Powerball and Mega Millions use a non-standard definition of odds. Odds are usually defined as the ratio of the positive result count to the negative result count. So the odds stated in the Powerball and Mega Millions websites should actually use one less than the number of combinations (292,201,337 and 258,890,849) since one combination is a positive result and the remainder are negative results. As a specific example, the Mega Millions odds table includes all the possible results. If the odds were stated with the standard definition, the probability of a negative result is equal to 
| Match | Prize | Odds | Probability = 1 - (Odds) / (Odds + 1) | Probability = 1 / Odds |
|---|---|---|---|---|
| Total | 0.688586303 | 0.999997720 |
||
| 5 + 0 | $1,000,000 | 18,492,203.5714 | 0.000000054 | 0.000000054 |
| 4 + 1 | $5,000 | 739,688.1429 | 0.000001352 | 0.000001352 |
| 4 + 0 | $500 | 52,834.8673 | 0.000018927 | 0.000018927 |
| 3 + 1 | $50 | 10,720.1180 | 0.000093274 | 0.000093283 |
| 3 + 0 | $5 | 765.7227 | 0.001304253 | 0.001305956 |
| 2 + 1 | $5 | 472.9464 | 0.002109943 | 0.002114405 |
| 2 + 0 | $0 | 33.7819 | 0.028750586 | 0.029601651 |
| 1 + 1 | $2 | 56.4712 | 0.017400019 | 0.017708141 |
| 1 + 0 | $0 | 4.0337 | 0.198661025 | 0.247911347 |
| 0 + 1 | $1 | 21.3906 | 0.044661599 | 0.046749507 |
| No Matches | $0 | 1.5279 | 0.395585268 | 0.654493095 |
The Megaplier odds are similar. The standard
Expected value without Multiplier or Jackpot
The non-jackpot prize amounts are fixed and the expected value is easy to calculate. As stated before, each prize amount is multiplied by the probably of happening and added together.
However, there is some variations in how taxes are handled. All lottery winnings are subject to a 39.6% federal tax. Most places, like New York, have a state income tax. Some cities, like New York City, also have a city tax (sucks to be me). Some other states, like California, have no income tax or no tax on lottery winnings (lucky to be you). Expected values for these two situations are included below. In addition, I include a third situation. When claiming your winnings, large prizes are paid directly by the state lottery office and taxes are partially deducted automatically. Small amounts are paid by any lottery retailer without any taxes deducted. I imagine some people might forget to declare these smaller winnings in their tax returns so I included two more columns for these situations.
Looking at the expected value, unsurprisingly, you can see that each dollar spent will return you less than one dollar. Interestingly, you can see that Mega Millions is a better value. Don't forget that Powerball is $2 per ticket, so we need to half the expected value to make the two values comparable.
| Mega Millions Tier | Odds | Prize | Pre-tax Expected Value | CA Post-tax Expected Value | NYC Post-tax Expected Value | CA Partial-tax Expected Value | NYC Partial-tax Expected Value |
|---|---|---|---|---|---|---|---|
| Total Expected Value | $0.1742 | $0.1052 | $0.0831 | $0.1501 | $0.0975 |
||
| 5 + 0 | 18,492,203.57 | $1,000,000.00 | $0.0541 | $0.0327 | $0.0258 | $0.0327 | $0.0258 |
| 4 + 1 | 739,688.14 | $5,000.00 | $0.0068 | $0.0041 | $0.0032 | $0.0041 | $0.0032 |
| 4 + 0 | 52,834.87 | $500.00 | $0.0095 | $0.0057 | $0.0045 | $0.0095 | $0.0057 |
| 3 + 1 | 10,720.12 | $50.00 | $0.0047 | $0.0028 | $0.0022 | $0.0047 | $0.0028 |
| 3 + 0 | 765.72 | $5.00 | $0.0065 | $0.0039 | $0.0031 | $0.0065 | $0.0039 |
| 2 + 1 | 472.95 | $5.00 | $0.0106 | $0.0064 | $0.0050 | $0.0106 | $0.0064 |
| 1 + 1 | 56.47 | $2.00 | $0.0354 | $0.0214 | $0.0169 | $0.0354 | $0.0214 |
| 0 + 1 | 21.39 | $1.00 | $0.0467 | $0.0282 | $0.0223 | $0.0467 | $0.0282 |
| Powerball Tier | Odds | Prize | Pre-tax Expected Value | CA Post-tax Expected Value | NYC Post-tax Expected Value | CA Partial-tax Expected Value | NYC Partial-tax Expected Value |
|---|---|---|---|---|---|---|---|
| Expected Value per Dollar | $0.1599 | $0.0966 | $0.0763 | $0.1322 | $0.0877 |
||
| 5 + 0 | 11,688,053.52 | $1,000,000.00 | $0.0856 | $0.0517 | $0.0408 | $0.0517 | $0.0408 |
| 4 + 1 | 913,129.18 | $50,000.00 | $0.0548 | $0.0331 | $0.0261 | $0.0331 | $0.0261 |
| 4 + 0 | 36,525.17 | $100.00 | $0.0027 | $0.0017 | $0.0013 | $0.0027 | $0.0017 |
| 3 + 1 | 14,494.11 | $100.00 | $0.0069 | $0.0042 | $0.0033 | $0.0069 | $0.0042 |
| 3 + 0 | 579.76 | $7.00 | $0.0121 | $0.0073 | $0.0058 | $0.0121 | $0.0073 |
| 2 + 1 | 701.33 | $7.00 | $0.0100 | $0.0060 | $0.0048 | $0.0100 | $0.0060 |
| 1 + 1 | 91.98 | $4.00 | $0.0435 | $0.0263 | $0.0207 | $0.0435 | $0.0263 |
| 0 + 1 | 38.32 | $4.00 | $0.1044 | $0.0630 | $0.0498 | $0.1044 | $0.0630 |
| Total Expected Value | $0.3199 | $0.1932 | $0.1526 | $0.2643 | $0.1754 |
Expected Value with multiplier without Jackpot
Both lotteries have a random prize multiplier that costs an extra $1 called the Powerplay and Megaplier. Intuitively, by spending only $1, you can increase your non-jackpot winnings by at least 2 times. So we should see increased expected value.
Each multiplier has some odds of being selected that week. So, using the same method above, we first calculate the expected values for each of the multipliers' and their prizes. The result of this calculation is given in the first column below. Then we calculate the expected value of expected values of multipliers, which is the second expected value column below. To make things a little more complicated, Powerplay includes a 10X multiplier when the jackpot is below $150M.
As you can see, the multipliers will increase your expected value but, unsurprisingly, not enough make money from the lottery. Again, Mega Millions remains a better value after adding the lotteries' multipliers. As before, since we are spending an extra dollar for the multiplier, we need to divide by the total cost of the ticket so all expected values are comparable.
| Megaplier | Pre-Tax Expected Value (similar to table above) | Odds | Pre-Tax Expected Value for this table |
|---|---|---|---|
| Expected Value per Dollar | $0.3368 |
||
| 2 | $0.3485 | 7.50 | $0.0465 |
| 3 | $0.5227 | 3.75 | $0.1394 |
| 4 | $0.6969 | 5 | $0.1394 |
| 5 | $0.8712 | 2.5 | $0.3485 |
| Total Expected Value | $0.6737 |
| Powerplay | Pre-Tax Expected Value (similar to table above) | Odds with 10X multiplier | Pre-Tax Expected Value with 10X | Odds without 10X multiplier | Pre-Tax Expected Value without 10X |
|---|---|---|---|---|---|
| Expected Value per Dollar | $0.2732 | $0.2597 |
|||
| 2 | $0.6398 | 1.7917 | $0.3571 | 1.7500 | $0.3656 |
| 3 | $0.8741 | 3.3077 | $0.2643 | 3.2308 | $0.2705 |
| 4 | $1.1084 | 14.3333 | $0.0773 | 14.0000 | $0.0792 |
| 5 | $1.3427 | 21.5000 | $0.0625 | 21.0000 | $0.0639 |
| 10 | $2.5143 | 43.0000 | $0.0585 | ||
| Total Expected Value | $0.8196 | $0.7792 |
Finally, the results for the same post tax situations as in the earlier section are given for completeness. These are all expected values per dollar.
| Tax Situation | Mega Millions w/o Megaplier | Mega Millions w/ Megaplier | Powerball w/o Powerplay | Powerplay w/ 10X multiplier | Powerplay w/o 10X multiplier |
|---|---|---|---|---|---|
| Pre-tax | $0.1742 | $0.3368 | $0.1599 | $0.2732 | $0.2597 |
| Post-tax CA | $0.1052 | $0.2035 | $0.0966 | $0.1650 | $0.1569 |
| Post-tax NYC | $0.0831 | $0.1607 | $0.0763 | $0.1303 | $0.1239 |
| Partial-tax CA | $0.1501 | $0.2903 | $0.1322 | $0.2306 | $0.2184 |
| Partial-tax NYC | $0.0975 | $0.2753 | $0.0877 | $0.2170 | $0.2051 |
Estimating Tickets Sold
We now have to estimate the expected value of the jackpot and simply add it to the numbers above. This can be tricky because, unlike before, having multiple winners will split our winnings. And to calculate the number of winners, we need to first estimate the number of tickets sold. Some places claim the number of tickets sold is half of the sales. I believe it is true that half of sales goes into the prize pool, but not all of that goes into the jackpot. We know how much the jackpot increases, but I don't know where to find overall sales. Florida, Texas, Colorado, New Mexico, and Virginia have FAQs which state that a maximum 34.0066% of sales are contributed towards the jackpot. In addition to not reaching that maximum, other states might have different maximums.
But there is a better way. Both Powerball and Mega Millions gives the details of how many winning tickets there were sold for each prize for each drawing. They even give the breakdown of winners with and without the $1 multiplier. So to estimate the number of tickets sold, we just multiply each type of winner by the probability of winning. Next, since we know the number of tickets sold with the $1 multiplier, we can estimate the total sales and then calculate the percent of sales that goes towards the jackpot. Once we know that percentage, it is easy to calculate the number of tickets sold for any Jackpot from the increase in its cash value.
For a specific example, lets take the $1.58B Powerball drawing. If we take the count of each prize winners multiplied by the probability of winning, we get nine estimates of the number of tickets sold, one for each tier of winner. But, looking at the number of tickets sold, using the number of jackpot winners gives an unreliable (high variance) estimate since there are so few winners. Similarly, looking at earlier weeks, there are sometimes very few (5+0) and (4+1) ticket winners. So I decided not use the top 3 tiers when looking at the estimates. This leaves us with six estimates of the number of tickets sold.
| Powerball 1/13/2016 Tier | Powerball winning tickets | Powerplay winning tickets | Odds of Winning | Approximate tickets sold |
|---|---|---|---|---|
| 5 + 1 | 3 | n/a | 292,201,338 | 876,604,014 |
| 5 + 0 | 73 | 8 | 11,688,053.52 | 946,732,335 |
| 4 + 1 | 827 | 107 | 913,129.18 | 852,862,654 |
| 4 + 0 | 20,544 | 2,834 | 36,525.17 | 853,885,424 |
| 3 + 1 | 47,685 | 6,597 | 14,494.11 | 786,769,279 |
| 3 + 0 | 1,164,124 | 157,552 | 579.76 | 766,254,878 |
| 2 + 1 | 895,097 | 120,695 | 701.33 | 712,405,403 |
| 1 + 1 | 6,343,237 | 840,981 | 91.98 | 660,804,372 |
| 0 + 1 | 14,595,721 | 1,914,561 | 38.32 | 632,674,006 |
You can see that the estimates are not the same for each tier and it is obvious to see that they increase as the probabilities get lower. However, looking at 11/14/2015, we can see that the estimates also vary but they decrease as the probabilities get lower. For 12/9/2015, there doesn't seem to be a trend with the estimates. In addition to this strange behavior, these six estimates should be much closer since we have so many winners. Specifically speaking, they are outside a 99% confidence interval based on binomial proportions. One possible reason for these anomalies is the assumption that all numbers are equally chosen. This is almost certainly not true since many people use dates to select numbers and others avoid the number 13. In the 1/13/2016 Powerball drawing, there are many low numbers which caused the increasing estimates as the probabilities get lower. So to handle this, we can look over more dates and the estimate of percent revenue to jackpot should average out.
| Powerball | 1/13/2016 Powerball tickets won | 1/13/2016 Powerplay tickets won | 1/13/2016 estimated tickets sold | 12/9/2015 Powerball tickets won | 12/9/2015 Powerplay tickets won | 12/9/2015 estimated tickets sold | 11/14/2015 Powerball tickets won | 11/14/2015 Powerplay tickets won | 11/14/2015 estimated tickets sold |
|---|---|---|---|---|---|---|---|---|---|
| % of Revenue into Jackpot | 25.36% | 30.09% | Ignored | ||||||
| 5 + 1 | 3 | N/A | 876,604,014 | 0 | N/A | 0 | 0 | N/A | 0 |
| 5 + 0 | 73 | 8 | 946,732,335 | 1 | 0 | 11,688,054 | 0 | 0 | 0 |
| 4 + 1 | 827 | 107 | 852,862,654 | 16 | 4 | 18,262,584 | 13 | 2 | 13,696,938 |
| 4 + 0 | 20,544 | 2,834 | 853,885,424 | 369 | 78 | 16,326,751 | 232 | 65 | 10,847,975 |
| 3 + 1 | 47,685 | 6,597 | 786,769,279 | 899 | 215 | 16,146,439 | 584 | 146 | 10,580,700 |
| 3 + 0 | 1,164,124 | 157,552 | 766,254,878 | 25,499 | 5,804 | 18,148,227 | 14,667 | 3,868 | 10,745,852 |
| 2 + 1 | 895,097 | 120,695 | 712,405,403 | 19,331 | 4,173 | 16,484,060 | 13,178 | 3,624 | 11,783,747 |
| 1 + 1 | 6,343,237 | 840,981 | 660,804,372 | 141,406 | 30,052 | 15,770,707 | 106,298 | 28,557 | 12,403,963 |
| 0 + 1 | 14,595,721 | 1,914,561 | 632,674,006 | 322,035 | 67,339 | 14,920,812 | 263,030 | 71,231 | 12,808,882 |
| Avg Tickets Sold | 735,465,560 | 16,299,499 | 11,528,520 |
||||||
| Percent Powerplay tickets sold | 11.66% | 17.44% | 21.26% | ||||||
| Total Revenue | $1,556,653,562.80 | $35,442,203.15 | $25,508,567.54 |
||||||
| Increase in Cash Value of Jackpot | $394,692,000 | $10,664,000 | Ignored |
% of Revenue into Jackpot
If we average the bottom six estimates of the 1/13/2016 drawing, this gives us an average tickets sold of 735,465,560. From the winners web page, we also know the ratio of Powerball to Powerplay tickets won which gives us an estimate for the ratio of tickets sold. Since Powerball tickets are $2 each and Powerplay tickets are $3 each, we now there was approximately $1,556,653,563 in revenue for this drawing. Since the cash value of the jackpot increased from $588M to $983M, This gives us about 25.36% of revenue into the jackpot for the 1/13/2016 drawing.
So lets go over more dates and average for a better estimate. To keep my life simple, I estimated from 10/7/2015 forward since the Powerball odds changed on that date. Unfortunately, there is still one last wrinkle. The minimum increase in the annuity jackpot is $10M. So when the jackpot increases the minimum amount, there is no relation between jackpot increase and tickets sold. For this reason I used only the drawing of 11/4/2015 and all the drawings from 12/9/2015 until 1/13/2016. Averaging all of the percent revenue to jackpot gives a result of 28.90%. This is a bit lower than the maximum of 34.0066% listed on a few lottery FAQs. So most states probably don't reach their maximum or have a lower maximum than 34.0066%. Lets take a look at Megamillions. There is a minimum increase of $5M for Megamillions. Using all the other drawings since 9/15/2015, the percent revenue to jackpot for Mega Millions is 26.63%
There are a few assumptions made here. We assume the states don't change how they are contributing to the jackpot. Even if they don't change, we can still run into problems if each state accounts for a different percent and states contribution proportion changes as the jackpot increases.
Calculating Expected Value with Current Jackpot
First, we need to estimate number of tickets sold for a specific drawing, it takes a few steps. First, to estimate the total revenues, take the increase in cash value from the previous drawing. If you only have the previous week's annuity value, you can convert by multiplying the annuity value by 0.62 to get the cash value. Next, divide by 28.9% or 26.63% for Powerball and Megamillions respectively. This gives us the total sales since the last drawing. Finally, We know that each Powerball ticket is $2, each Powerplay ticket is $3 or each Megamillions ticket is $1, each Megaplier ticket is $2. We just need to estimate the ratio of ticket types and we can find out how many tickets were sold in total.
For the 1/13/2016 drawing, 11.6% of the tickets won were Powerplay tickets. However, for smaller jackpots, as much as 21.4% of the tickets won were Powerplay tickets.For the recent drawings, the percent of Megaplier tickets only ranged from 10% to 12.5% but this didn't include any of the large Mega Millions jackpots. Choose an appropriate percent of Powerplay/Megaplier tickets based on the jackpot size using the section below as guidance.
Finally, the number of tickets can be calculated as
And once we estimate the number of tickets sold, we can calculate the probability of having a different number of winners and then calculate the expected value of the jackpot. For the big 1/13/2016 drawing of Powerball, the details are worked out below assuming 735,465,560 tickets sold and $594,075,072 cash value jackpot after applying only 39.6% federal income tax. For the 1/13/2016 Powerball jackpot, a $2 ticket had an expected value per dollar of $0.3713. When combining this with the non-jackpot prizes, we have a total of $0.4679 per dollar, less than the dollar that we spent. For a $3 powerplay ticket, we have an expected value per dollar of $0.2475 for the jackpot. After combining with the non-jackpot prizes, we get an expected value of $0.4044 per dollar, also less than the dollar we spent. This says, considering a large jackpot, we are better off not spending the extra dollar on power play. This is in contrast to when we didn't consider any jackpot at all.
| Number of Winners | Probability | Assuming I win, how much will I get after splitting with other winners? | Assuming I win, what is my expected value? | |
|---|---|---|---|---|
| Expected Value per Dollar | $0.2932 | |||
| 0 | 8.07% | 469,201,278.72 | $37,865,851.28 | |
| 1 | 20.31% | 234,600,639.36 | $47,653,836.57 | |
| 2 | 25.56% | 156,400,426.24 | $39,981,285.86 | |
| 3 | 21.45% | 117,300,319.68 | $25,158,046.01 | |
| 4 | 13.50% | 93,840,255.74 | $12,664,470.67 | |
| 5 | 6.79% | 78,200,213.12 | $5,312,707.82 | |
| 6 | 2.85% | 67,028,754.10 | $1,910,284.41 | |
| 7 | 1.02% | 58,650,159.84 | $601,018.97 | |
| 8 | 0.32% | 52,133,475.41 | $168,083.78 | |
| 9 | 0.09% | 46,920,127.87 | $42,306.39 | |
| Total Expected Value | $171,357,891.76 | |||
| Expected Value of $2 ticket (not assuming I win) | $0.5864 |
On 1/8/2016, there was a winner of a $165M Mega Millions annuity jackpot. After converting to cash value and taking out only federal income tax, we are left with $61,789,200.00. Using an estimated 41,075,760 tickets sold, we get an expected value of $0.2207 per dollar for a combined total of $0.3259 for a $1 Mega Millions ticket. For a $2 ticket with the megaplier, we get $0.3138 per dollar. Here, we are also better of not spending the additional $1 for larger jackpots and instead of spending the extra money on more tickets.
Below are plots for the most recent jackpot Powerball and Mega Millions wins. The plots contain the growth of the jackpot along with the growth of expected value. We can see that expected value continued to climb even as the jackpot climbs. Hopefully, one day we will see a jackpot that would give a higher expected value than the cost.
Powerplay and Megaplier
One more important detail is how many people buy the $1 multipliers as the Jackpot increases. It looks like the regular players like to buy the $1 multipliers for both lotteries. But as the jackpot increases, people would rather buy more tickets for a better chance at the jackpot.
While the Powerball curve below includes a very large jackpot, the Megamillions plot only has medium sized jackpots. For Megamillions, I looked at some of the recent large jackpots. The largest jackpot on 3/30/2012 had 5.3% Megaplier purchased. The second largest jackpot on 12/18/10213 had 7.4%, and the most recent large jackpot on 3/18/2014 was 7.7%.
