Computing the Lottery Odds

How MegaMillions and Powerball Figure the Jackpot

Feb 20, 2008 James Hutchinson

How lotteries compute the odds of winning, and a comparison of the odds of MegaMillions and Powerball.

In the United States, MegaMillions and Powerball are large, multi-state lotteries that offer huge payouts if the lucky winner can select all the numbers exactly.

The jackpots change each drawing, rising every time there is no winner, and also can be adjusted based on an expected increase in sales. Sales do go up when there is a large jackpot.

Although the jackpots change, the odds on winning the big prize do not. They are fixed based on how the lottery is set up, specifically how many balls are drawn and the total number of possible numbers.

MegaMillions and Powerball are very similar in set up, but have slight differences, which make a difference in the odds. The details below will show how to compute the odds on the large jackpots, and the same methodology can be applied to smaller payouts or other lotteries.

Calculating MegaMillions Payout

The odds of winning the jackpot in the MegaMillions drawing are 175,711,536 to 1. The required data are:

  • Five numbers are chosen from a pool of 56 numbers, once the numbers are drawn they are not returned to the pool, therefore a number can only be drawn once from this pool.
  • One number is chosen from a separate pool of 46 numbers for the red ball.

To compute the odds, statistically the chances of getting the first number right are five chances out of 56. Any one of the five numbers on the card will do, and the order is not important.

With the next ball, the chances are now four out of the 55 remaining balls. Continuing, three out of 54, two out of 53, and then there is only one chance that the final needed number out of the first pool will be the chosen out the 52 remaining.

The odds for the first five numbers are multiplied, (not added) together, and then the result is multiplied by the one chance out of 46 in the separate pool

Mathematically:

5/56 times 4/55 times 3/54 times 2/53 times 1/52 times 1/46 = 1 out of 175,711,536.

Calculating Powerball Payout

The formula for the Powerball jackpot is the same, with only the numbers changing. With five numbers chosen out of a pool of 55 balls, and one number from the separate pool of 42, the odds are 146,107,962 to one.

The chances of winning Powerball are slightly higher than MegaMillions, but that generally means smaller jackpots. Realistically, the odds are so high that any difference is negligible for an individual ticket.

Other Differences between MegaMillions and Powerball

In addition to number of balls, the implied interest rates for the cash payout are different, with Powerball currently using a higher rate, and therefore a lower cash option. For the payments option MegaMillions pays over 26 years, and Powerball over 30.

The copyright of the article Computing the Lottery Odds in Consumer Education is owned by James Hutchinson. Permission to republish Computing the Lottery Odds in print or online must be granted by the author in writing.
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Comments

Dec 9, 2008 11:36 AM
Guest :
This is all a lie, the chances are much, much worse and its because you are following a failed mathematical logic.

Ball one - 1 ball out of 56 (it can be any ball)
ball two - 1 ball out of 55 (it can be any ball)
ball three - 1 ball out of 54 (can be any ball)
ball four - 1 ball out of 53 (any ball)
ball five - 1 ball out of 52 (any ball again)
Yellow ball - 1 ball out of 46 (separate pool)

56x55x54x53x52x46= appx 21 billion/1
Dec 9, 2008 11:44 AM
James Hutchinson :
For the comment, that is not correct. Since ball one can be any of the 5, the odds on getting any of the 6 balls are 5/56, not 1/56.
If it was 21 billion to 1, no one would ever win.
Thanks for reading.
Dec 14, 2008 11:49 AM
Guest :
I agree with Guest, the true odds are 21 billion to 1. Computing odds boils down to one synopsis: how many different possible combinations exist from which one combination must match. The Guest's formula of 56x55x54x53x52x46 is absolutely correct.

On a much smaller scale, if you were asked to name two cards chosen randomly from a standard deck of cards, there would be 2,652 total possibilities (52x51), therefore, the odds would be 2,652 to 1 against naming them correctly—before the draw. If the first card was a match, then yes, the odds of naming the second card drop to 51 to 1, but not if you must name both cards before any are drawn, then the odds remain 2,652 to 1 for the entire draw process.

I believe the writer goes awry in assuming that the first number drawn has already been matched, in that case, the odds of correctly naming the second number do drop significantly, and the odds would drop with each subsequent number being matched, however, as in any lottery game, ALL the numbers must be chosen before any ball is drawn thereby rendering meaningless any consideration of subsequent drawn numbers.

As far as no one ever winning, that's already been proven wrong via all the past winners. Whenever the jackpot passes, or approaches, $100 million, it's been shown that the number of tickets purchased goes up dramatically, therefore, the odds of someone matching the numbers goes down because of the additional millions of tickets being sold.

My question is: does anyone really know where the money goes—as opposed to where [lying] politicians say it goes; it seems like today's kids have become a hell of a lot stupider since lottery money has been [supposedly] financing education. In the 1940s, 1950s, and 1960s, the USA was #1 in everything, today, educationally, we're #18 out of 24, and just one viewing of Jay Leno's "JayWalking" will convince anyone that there's too many idiots walking our streets.
Dec 14, 2008 12:48 PM
James Hutchinson :
I do maintain the odds are what I quoted in the article. As a proof, consider what the odds are on getting 1 right out of 5 choices. The odds on that first ball are not 1 out of 56, it's 5 of out 56.

The mistake the commenters are making is that the numbers do not have to be drawn in the exact order. If they were, then their supposition would be correct. But they don't so that the odds on the first number are not 1 out of 56, but 5 out of 56.

This may be a case of a political bias blinding the person's understanding of the mathematical concept. Yes, more people buy tickets when the jackpot is higher, but more tickets bought don't change the odds on winning. Each ticket still has the same chance.

As for the 21 billion to one, that would mean that every person on earth would have to buy 3 tickets to make it a one to one odds against the lottery, or every American would have to buy 80. Think about it.
Jan 2, 2009 8:28 AM
Guest :
Of'course James is correct.

You need to factor in the fact that the order of the picked numbers does not matter..

Jan 12, 2009 2:37 PM
Guest :
Ball 1: 5 total chances to match
Ball 2: 4 total chances to match
Ball 3: 3 total chances to match
Ball 4: 2 total chances to match
Ball 5: 1 chance to match

Equation (56/5)(55/4)(54/3)(53/2)(52/1) = 3,819,816/1
the odds of winning are 3,819816 to 1!

As far naming two cards from a standard deck: (52/2)(51/1) = 1326 to 1

JC
Feb 8, 2009 11:47 PM
Guest :
Wow 21 billion thats only 5 Billion away from the entire population on earth including infants that can't purchase lottery tickets LOL.
Apr 16, 2009 5:11 AM
Guest :
Handle: Pliny
---------------

Of course, Mr. Hutchinson is absolutely correct in his mathematics. This is a *combination* problem, not a permutation. In other words, no one should care at all about the different ways the particular numbers can be ordered. It's totally irrelevant to anyone who plays this game.

Anyway, just use intuitive ability and plain old common sense. No one is going to play this game if the odds of winning are in the billions. You might as well flush your money down the toilet.
Jun 17, 2009 10:18 PM
Guest :
I haven't checked the math but the logic used by the article to calculate the odds is correct. The 21 bill guest #1 gets has to be divided by how many ways you can permute/arrange(sp) the 5 regular numbers since like others have said the order in which you get those 5 numbers do not matter.
21 bill divided by (5*4*3*2*1) is about 175.something mill. (math checks out)

If you don't divided by the number of ways to arrange and get the odds of 21 bill you are assuming of the 5 regular numbers order matters. That is, 1,2,3,4,5 is NOT THE SAME as 5,4,3,2,1 2,3,4,5,1 4,1,2,3,5 and any other way you can arrange them. But the lottery uses those combinations interchangeably.

21 billion would mean you have to get the numbers AND the positions of those numbers correct.

Don't know if that helps any. Also sorry if I have typos or bad grammar. It's late and I'm tired and unemployed and the list of awfulness goes on.
Jun 17, 2009 10:27 PM
Guest :
I think the odds for the powerball have changed. It's draw 5 from a pool of 59 and 1 from 39 now.
10 Comments