Columnist; Science and Academics
Mega Millions shares many parallels with Powerball, including a rolling jackpot and a similar number matrix. The difference is that in Mega Millions, there are no add-ons to choose from, and the prizes are assigned by random multipliers determined at the time of the purchase.
Category
Mega Millions is a combinatorial lottery game drawing from two independent pools: Five numbers drawn from 1 – 70 (the base draw) and one number drawn from 1– 24 (the Mega Ball).
Size of the draw: 6, that is, 5 + 1.
Size of a line: 6, that is, 5 + 1.
Cost of a line: $5.
There are 5 multipliers (2x, 3x, 4x, 5x, 10x), of which one is drawn automatically for each player (on their ticket) before the draw.
Prizes
There are nine prize tiers, defined as matching 1 to 6 numbers in the draw (1 for the Mega Ball). They are all fixed prizes, except the top prize (6 numbers matched), which is a rolling jackpot.
The prizes do not increase uniformly from the lowest to the highest. There are abrupt jumps: from the third tier to the second tier and from the fourth tier to the 3rd tier, as well as the corresponding odds of winning.
Odds of winning
The odds of winning after the multiplier draw (after the purchase) range from 1 in 290,472,336 (for the jackpot; one of the lowest winning probabilities across lottery games and across all known games of chance, similar to Powerball) to 1 in 35 (for the lowest prize). The overall odds of winning a prize are 1 in 23.
The table below shows the odds of winning each possible prize, in two different probability fields (corresponding to two moments – before the multiplier draw and after it).
| Prize Amount | Ways to Win | Odds Before Multiplier Draw | Odds After Multiplier Draw |
|---|---|---|---|
|
Jackpot |
Match 5 + 1 |
1 in 290,472,336 |
1 in 290,472,336 |
|
$10,000,000 |
Match 5 and 10× |
1 in 12,629,232 |
1 in 404,135,424 |
|
$5,000,000 |
Match 5 and 5× |
1 in 12,629,232 |
1 in 202,067,712 |
|
$4,000,000 |
Match 5 and 4× |
1 in 12,629,232 |
1 in 101,033,856 |
|
$3,000,000 |
Match 5 and 3× |
1 in 12,629,232 |
1 in 40,413,542 |
|
$2,000,000 |
Match 5 and 2× |
1 in 12,629,232 |
1 in 26,900,265 |
|
$100,000 |
Match 4 + 1 and 10× |
1 in 893,761 |
1 in 28,600,352 |
|
$50,000 |
Match 4 + 1 and 5× |
1 in 893,761 |
1 in 14,300,176 |
|
$40,000 |
Match 4 + 1 and 4× |
1 in 893,761 |
1 in 7,150,088 |
|
$30,000 |
Match 4 + 1 and 3× |
1 in 893,761 |
1 in 2,860,035 |
|
$20,000 |
Match 4 + 1 and 2× |
1 in 893,761 |
1 in 1,966,274 |
|
$5,000 |
Match 4 and 10× |
1 in 38,859 |
1 in 1,243,488 |
|
$2,500 |
Match 4 and 5× |
1 in 38,859 |
1 in 621,744 |
|
$2,000 |
Match 4 and 4× OR Match 3 + 1 and 10× |
1 in 10,222 |
1 in 183,335 |
|
$1,500 |
Match 4 and 3× |
1 in 38,859 |
1 in 124,349 |
|
$1,000 |
Match 4 and 2× OR Match 3 + 1 and 5× |
1 in 624 |
1 in 60,397 |
|
$800 |
Match 3 + 1 and 4× |
1 in 13,965 |
1 in 111,720 |
|
$600 |
Match 3 + 1 and 3× |
1 in 13,965 |
1 in 44,688 |
|
$400 |
Match 3 + 1 and 2× |
1 in 13,965 |
1 in 29,745 |
|
$100 |
Match 3 and 10× OR Match 2 + 1 and 10× |
1 in 317 |
1 in 10,155 |
|
$70 |
Match 1 + 1 and 10× |
1 in 86 |
1 in 2,752 |
|
$50 |
Match 3 and 5× OR Match 2 + 1 and 5× OR Match Mega Ball only and 10× |
1 in 32 |
1 in 918 |
|
$40 |
Match 3 and 4× OR Match 2 + 1 and 4× |
1 in 317 |
1 in 2,539 |
|
$35 |
Match 1 + 1 and 5× |
1 in 86 |
1 in 1,376 |
|
$30 |
Match 3 and 3× OR Match 2 + 1 and 3× |
1 in 318 |
1 in 1,015 |
|
$28 |
Match 1 + 1 and 4× |
1 in 86 |
1 in 688 |
|
$25 |
Match Mega Ball only and 5× |
1 in 35 |
1 in 560 |
|
$21 |
Match 1 + 1 and 3× |
1 in 86 |
1 in 275 |
|
$20 |
Match 3 and 2× OR 2 + 1 and 2× OR Match Mega Ball only and 4× |
1 in 32 |
1 in 198 |
|
$15 |
Match Mega Ball only and 3× |
1 in 35 |
1 in 112 |
|
$14 |
Match 1 + 1 and 2× |
1 in 86 |
1 in 183 |
|
$10 |
Match Mega Ball only and 2× |
1 in 35 |
1 in 75 |
Average win and fairness
The next table shows the average win (or return) for each prize amount:
| Prize Amount | Average Win |
|---|---|
|
Jackpot |
Jackpot ÷ 290,472,336 |
|
$10,000,000 |
$0.02474 |
|
$5,000,000 |
$0.02474 |
|
$4,000,000 |
$0.03959 |
|
$3,000,000 |
$0.07423 |
|
$2,000,000 |
$0.07435 |
|
$100,000 |
$0.00350 |
|
$50,000 |
$0.00350 |
|
$40,000 |
$0.00559 |
|
$30,000 |
$0.01049 |
|
$20,000 |
$0.01051 |
|
$5,000 |
$0.00402 |
|
$2,500 |
$0.00402 |
|
$2,000 |
$0.01091 |
|
$1,500 |
$0.01206 |
|
$1,000 |
$0.01656 |
|
$800 |
$0.00716 |
|
$600 |
$0.01343 |
|
$400 |
$0.01345 |
|
$100 |
$0.00985 |
|
$70 |
$0.02544 |
|
$50 |
$0.05447 |
|
$40 |
$0.01575 |
|
$35 |
$0.02544 |
|
$30 |
$0.02956 |
|
$28 |
$0.04070 |
|
$25 |
$0.04464 |
|
$21 |
$0.07636 |
|
$20 |
$0.10101 |
|
$15 |
$0.13393 |
|
$14 |
$0.07650 |
|
$10 |
$0.13333 |
For an average jackpot won of $245 million*, the average win per line is $0.843.
For example, a $5,000 win (Match 4 prize tier) is distributed as an average win per line of $0.12867 (assuming the prize is won with a frequency of about 1 in 38,859 draws) over the long run.
The next table below shows the fairness of each prize relative to its probability:
| Prize Amount | Fairness |
|---|---|
|
$10,000,000 |
0.19% |
|
$5,000,000 |
0.09% |
|
$4,000,000 |
0.15% |
|
$3,000,000 |
0.29% |
|
$2,000,000 |
0.29% |
|
$100,000 |
0.07% |
|
$50,000 |
0.03% |
|
$40,000 |
0.05% |
|
$30,000 |
0.10% |
|
$20,000 |
0.10% |
|
$5,000 |
0.08% |
|
$2,500 |
0.08% |
|
$2,000 |
0.27% |
|
$1,500 |
0.24% |
|
$1,000 |
0.32% |
|
$800 |
0.17% |
|
$600 |
0.33% |
|
$400 |
0.33% |
|
$100 |
0.19% |
|
$70 |
0.87% |
|
$50 |
0.98% |
|
$40 |
0.39% |
|
$35 |
0.50% |
|
$30 |
0.49% |
|
$28 |
0.72% |
|
$25 |
0.71% |
|
$21 |
1.53% |
|
$20 |
1.53% |
|
$15 |
3.06% |
|
$14 |
1.43% |
|
$10 |
6.04% |
For an average jackpot of $245 millions*, its fairness would be 16.86%. However, this only applies in the circumstance where such a jackpot is actually won, as rollovers actually mean zero prizes in this tier.
The highest fairness is associated with the prizes of $10 and $15.The rest of the prizes have very low fairness (most under 1%), and many are lower than in Powerball; however, fairness was estimated per prize amount rather than per tier. A certain range of prize amounts per tier (determined by the various multipliers) will be fairer than individual amounts.
Expected value and house edge
We cannot compute a precise expected value of Mega Millions, as the first-tier prize is variable. However, we can estimate it in two different ways, associated with two assumptions:
- A: The jackpot amount is counted as zero (not won).
- B: The jackpot amount is taken as the average jackpot won over a period of time*.
In case A, the expected value is –$3.88, or –77.60% as a percentage of the line cost. The house edge of the game is 77.60%.
In case B, the expected value is –$3.04, or –60.80% as a percentage of the line cost. The house edge of the game is 60.80%.
One way to interpret this is that in case A, you are expected to lose on average $3.88 of every $5 wagered, over the long run.
*Note: The average jackpot won was chosen as an estimate for the last 10 years, using the data from the official Mega Millions website.
Final remarks
Like Powerball, Mega Millions is best suited for players chasing big wins, including the jackpot, rather than for those aiming for a positive balance over the long run, including by playing in groups or lottery syndicates.
The odds of winning lower prizes are still low, and such prizes have poor fairness, while the house edge is high in both games. Considering the cost per line and house edge of Powerball are far lower than in Megamillions, and the magnitude of the odds is about the same in the two games, Powerball is preferable for long-run players chasing the jackpot. For long-run players aiming for a positive balance, Megamillions is preferable due to its lower house edge when the jackpot is excluded.
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