I. Motivation

So, is there any STRATEGY at all for this RANDOM game? Our weapon to deal with this question is MATH, and nothing other than that. Just a bit of INTUITION is a flavor.

II. Bayesian Inference

As with the Hat Problem or the Monty Hall Problem, we can maximize the probability of winning a prize by discarding less likely cases where three or more consecutive numbers appear in the game balls. Theoretically speaking, the probability of the events that three or more consecutive numbers appear in the same draw is exactly the same as that of the case for any three or more numbers to appear, but the number of combinations is much less than the rest, so we seldom see the case in the real world. There were only 2 times that have occurred during the entire Mega Millions history. Suppose you are to bet on either of the following two cases: the next drawn numbers will include three or more consecutive numbers, or they will not. OK, you learned from your experience that the second case is the obvious way to go. With this simple math, you just have eliminated 54x₅₃C₂=74412 combinations from your choices.

III. Mega Millions Draw Numbers Analysis

We also take the Law of Large Numbers into account to calculate the optimal set of coefficient for each number. That is, if the number "1" appeared only once whereas the number "56" appeared 99 times in 100 draws, then you would suspect the probability of appearance of each number is equal. The probability of this event is ₁₀₀C₁(5/56)⁹⁹(51/56)^100-99 = 1/(8x10¹⁰²), which is far lower than the chances to win a Jackpot. This tells us that some numbers that have not appeared as much as others so far should have more "tendency" to appear in the future. In other words, the normalized standard deviation of the number of appearance should decrease as time elapses. Details can be found at the Statistics for the Mega Millions Winning Numbers. We can also consider "windowing" the appearances for certain period. In average, each gameball should appear once in 56 trials and each megaball should appear once in 46 trials. However, you will find some numbers have not appeared in more than the last 200 draws (x 5 picks/draw = 1000 picks) for 2 years, which by itself is a very strange result from the unbiased uniform-distribution point of view. This may mean not all the balls are uniform - in terms of shape, size, material, weight, and axis. However, if the balls are changed to new ones once in a while, you may want to think these numbers have strong potential to be drawn soon.

IV. Expectation of Winning Prizes

The odds of winning prizes are shown below. This table is brought from the official Mega Millions website.

From the table, you can calculate the cumulative probability of winning a prize and the expectation of the prize.

- If you bought 1 ticket

You have 1/75 + 1/141 + 1/844 + 1/306 + 1/13,781 + 1/15,313 + 1/689,065 + 1/3,904,701 + 1/175,711,536 = 1/40 chances to win a prize with the expectation of $2/75 + $3/141 + $10/844 + $7/306 + $150/13,781 + $150/15,313 + $10,000/689,065 + $250,000/3,904,701 + Jackpot/175,711,536 = $0.83. Here we assumed Jackpot prize is $12,000,000. This can be interpreted statistically as follows: You pay $1 to buy 1 ticket, and you have 2.5% of chances to get $0.83 back out of your $1 and 97.5% of chances to lose your $1. So, the expected net worth of the 1 ticket is 0.025x$0.83=$0.02, which is about 2% of the purchased ticket price. In case the Jackpot is rolled over, the expectation of the prize will increase, but increasing the Jackpot prize from $12,000,000 to $120,000,000 only results in increasing the expectation of the prize from $0.83 to $1.44, thus the expected net worth of a ticket increases from 0.025x$0.83=$0.02 to 0.025x$1.44=$0.036, which is a trivial difference.

- If you bought 2 tickets

The chances for at least one of 2 tickets to win a prize is 1-(39/40)² =0.049 with the expectation of $1.65. So, you pay $2 to buy 2 tickets, and you have 4.9% of chances to get $1.65 back out of your $2 and 95.1% of chances to lose your $2. The expected net worth of the 2 tickets is 0.049x$1.65=$0.08, which is about 4% of the purchased tickets price.

- If you bought 40 tickets

The chances for at least one of 40 tickets to win a prize is 1-(39/40)⁴⁰ =0.637 with the expectation of $33.04. So, you pay $40 to buy 40 tickets, and you have 63.7% of chances to get $33.04 back out of your $40 and 36.3% of chances to lose your $40. The expected net worth of the 40 tickets is 0.637x$33.04=$21.05, which is about 52% of the purchased tickets price.

- If you bought 100 tickets

The chances for at least one of 100 tickets to win a prize is 1-(39/40)¹⁰⁰ =0.92 with the expectation of $83. So, you pay $100 to buy 100 tickets, and you have 92% of chances to get $83 back out of your $100 and 8% of chances to lose your $100. The expected net worth of the 100 tickets is 0.92x$83=$76.4, which is about 76% of the purchased tickets price.

So, the more tickets you buy, the chances of at least one of your tickets win a prize increase with higher expected amount of prize, and thus the chances of your getting nothing decrease, but the amount of money you lose increases when you do lose. The percentage of the expectected net worth of the tickets gradually converges to the purchased tickets price, at the risk of losing more money with less probability.

V. Conclusions

Consequently, it is obvious that you are playing a game that is not in favor of your side if you don't have any strategy to tackle this randomness. Just randomly generated numbers are of no help for you to increase your chances to win a prize.

Fortunately, we came up with a solution that could help you to increase your chances. We generate a pool of balls while weighting the number of each ball differently, and then pick balls randomly from the pool. Finally, you are convinced to check out our Mega Millions Winning Numbers Generator. You can also use it on your Mobile Phones. Good luck!