How to Calculate Mega Millions Lottery Odds

For any lottery, the odds of winning a prize depend on the number of lottery ticket combinations. The Powerball and Mega Millions lotteries have a similar structure, with five regular balls and one special ball. With a little math, you can easily calculate the chances of holding the winning ticket.

In the Mega Millions lottery, the five regular white balls have numbers that range from 1 to 56. The special ball, the "Mega Ball," has a number that ranges from 1 to 46. The five regular numbers must be distinct from one another, but sixth number may be equal to one of the other five. You win the lottery jackpot when you match all six numbers. There are smaller prizes for matching only some of the numbers. Here is a breakdown of the Mega Millions lottery and the odds of winning each kind of prize.

All Six Numbers--The Jackpot

The total number of Mega Millions ticket combinations is

  56 x 55 x 54 x 53 x 52
------------------------------ x 46 = 175,711,536
       5 x 4 x 3 x 2 x 1

Therefore, the odds of winning the jackpot are 1 in 175,711,536, or 0.0000000056911.

All Five White Balls--$250,000

For every winning Mega Millions jackpot ticket, there are 45 other tickets that match the five white balls, but not the the Mega Ball. Thus, the odds of winning the $250,000 lottery prize are 45 out of 175,711,536, or 1 out of 3,904,700.8. This is 0.0000002561.

Four White Balls + The Mega Ball--$10,000

For every winning jackpot ticket, there are 255 other ticket combinations that have one white ball incorrect. How so? For each of the 5 correct balls, you can replace it with one of 51 incorrect balls. Since 5 x 51 = 255, the odds are 255 in 175,711,536 of winning $10,000. This is equivalent to 1 out of 689,064.85, or 0.0000014512.

Four White Balls--$150

There are 255 x 45 = 10,260 ways to match only 4 white balls, thus the odds of winning this prize are 11,475 out of 175,711,536. This equivalent to 1 in 15,312.55, or 0.000065306.

Three White Balls + The Mega Ball--$150

How many ways can you take 2 of the 5 correct white balls and replace them with 2 out of 51 incorrect white balls? The answer is

 5 x 4       51 x 50
-------- x ------------ = 12,750
 2 x 1         2 x 1

Since 12,750 out of 175,711,536 is equivalent to 1 in 13,781.3, the total probability is 0.000072562.

Three White Balls--$7

There are 12,750 x 45 = 573,750 ways to match only 3 white balls. This works out to 1 in 306.25, or 0.0032653.

Two White Balls + The Mega Ball--$10

We must count the number of ways to take 3 out of 5 correct white balls and replace them with 3 out of 51 incorrect balls. This is

 5 x 4 x 3      51 x 50 x 49
------------ x ----------------- = 208,250
 3 x 2 x 1         3 x 2 x 1

This is equivalent to 1 out of 843.75, or 0.0011852.

One White Ball + The Mega Ball--$3

The number of ways to get 1 white ball correct and 4 incorrect is

  51 x 50 x 49 x 48
------------------------ x 5 = 1,249,500
      4 x 3 x 2 x 1

Thus the odds of winning the $3 prize are 1 in 140.63, or 0.0071111.

Just the Mega Ball--$2

The number of ways to pick 5 incorrect white balls and the correct power ball is

  51 x 50 x 49 x 48 x 47
------------------------------ = 2,349,060
       5 x 4 x 3 x 2 x 1

This is 1 out of 74.8, or 0.013369. Notice that the odds are not 1 out of 46. This is because we are counting a more specific quantity: the number of ways of getting the Mega Ball right and the other five balls wrong. If we are computing just the probability of getting the Mega Ball right, and ignoring the white balls, it would be 1 out of 46.

Also note that there is no prize for getting only one or two white balls correct.

© Had2Know 2010

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