Lottery Odds

We all dream of winning the lottery, but what are the lottery odds?

This resource guide will you help you calculate lottery odds or probability. To skip all the background about probability theory, you can go directly to the table below to find out which lottos provide the best odds.

So how does one calculate lotto odds?

Understand the lottery rules

Before participating in any game, make sure you find out the rules of the game. To win, players need to select a winning combination of numbers. Several formats exist; however the most common is for players to select k numbers from a pool of numbers ranging from 1 to n. To win the jackpot prize, you need to match all k numbers of the winning combination. A popular format of this game is Lotto 6/49 offered in Canada and in the UK where 6 numbers out of a set of 49 is selected. A variation of this game is to have players to also select number(s) from a second set of numbers. This variation is used in many lotteries around the world including US Powerball, Mega Millions, and Euro Millions.

Understand probability concepts applied in lotteries

In lottery mathematics, you will come across the concept of permutations and combinations. Many confuse the two, but the main difference is that permutations consider order (of the numbers drawn) to be important, while in combinations, order doesn’t matter. Most lotteries ignore the order of winning numbers, so you should generally be concerned with combinations.

Lottery Probability | Lotto Ticketmaster

This calculation can be performed in Microsoft Excel using the combination function. This function is implemented as COMBIN(n,k).

Going back to our example of Lotto 6/49,  we have n=49, the total number of possible choices. k=6, the number of choices players can make. So the calculation would be:

Lotto 6/49 Odds | Lotto Ticketmaster

So the total number of possible combinations, or odds, for any 6 digit number to win the Lotto 6/49 jackpot would be 13,983,816. The reciprocal of this number gives the probability of 0.0000000715 or 0.00000715%.

To get the odds of other possibilities, the general formula is known as the hypergeometric distribution:

Lottery Odds | Lotto Ticketmaster

where N= number of balls in lottery

mn = number of balls in a single ticket

k is the number of matching balls for a winning ticket

For example to determine the odds of matching 5 numbers in a 6/49 draw, the expression would be:

6/49 Lottery Odds | Lotto Ticketmaster

Solving this expression yields an answer of 2,330,636 possible combinations.

Calculations can easily be performed in popular spreadsheets using the HYPGEODIST() function.

Bonus Balls

Many lotteries including Powerball and Mega Millions have a “bonus ball”. If this bonus ball is drawn from a second set of numbers, then simply multiply the odds by the number of balls in the second set.

When more than one bonus ball is drawn, as in the Euro Millions game, treat the bonus balls as a mini-lottery in their own right. Simply multiply the odds of this mini-lottery with the odds of matching the first set of numbers.

Review the lotto jackpot odds

We built a table showing the odds of winning the jackpot for many lottery draws in the world. The winning combination of numbers is based on matching n numbers out of a set of k numbers. Some lotteries also require x numbers out of a second set of y numbers.

LotteryJackpot Winning CombinationOdds
Lotto 6/49Match 6/49 1 in 13,983,816
US PowerballMatch 5/69 + 1/261 in 292,201,338
US Mega MillionsMatch 5/75 + 1/151 in 258,890,850
EuroMillionsMatch 5/50 + 2/111 in 116,531,800
SuperEna LottoMatch 6/901 ini 622,614,630
Uk LotteryMatch 6/491 in 13,983,816
German LotteryMatch 6/49 + 1/101 in 139,838,160
France LotoMatch 5/49 + 1/101 in 19,068,840
Mega SenaMatch 6/601 in 50,063,860
El GordoMatch 5/54 + 1/101 in 31,625,100
Australian PowerballMatch 6/40 + 1/201 in 76,767,600
Euro JackpotMatch 5/50 + 2/101 in 95,344,200
La PrimitivaMatch 6/491 in 13,983,816
SuperLotto PlusMatch 5/47 + 1/271 in 41,416,353

Select winning numbers

Each number is as equally likely to come up in the draw as the next number. As each number has as much of a chance to come up as the other, you can certainly base your selection of numbers on birthdays, lucky numbers, anniversaries, etc. There are no secret strategies for picking winning numbers to improve odds. A seemingly improbable combination of 1-2-3-4-5-6 is no more or less likely to win as a more random looking combination of 34-32-5-17-3-49. However, check our Select Winning Lotto Numbers guide to help improve not necessarily the odds, but the amount of money you can win.

Odds are not as bad as you might think

NOAA puts the odds of being killed by lightning in the US in any given year at 1 in 1.9m. So does this mean you have a better chance to be struck by lighting than to win a lottery jackpot? How about the odds of other crazy things more likely to happen as shown in the video below?

Lets put things in perspective by looking at some statistics. In 1996, the number of people winning over $1,000,000 playing North American lotteries was 1,136. Moreover, an additional 4,520 people won $100,000 or more. By contrast, 91 people were killed by lightning.1  Furthermore, the number of deaths caused by lightning has been dropping through the years, whilst the number of people winning million dollar jackpots has been rising.

Also consider there is no second prize in a lighting strike (although surviving one might be considered as one). With a lottery, you can win smaller cash prizes by matching some of the winning numbers. North American lotteries award over $50,000,000 in prizes every day. Even lightning is not that striking!

Remember that every single lottery winner is just an average person, like yourself. Keep in mind that no one has ever won without buying a ticket! So take a chance and imagine how your lifestyle can change if you win one of these upcoming jackpot draws.

 

Have fun while responsibly playing your favourite lottery!

North American Association of State and Provincial Lotteries

 

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