Lottery mathematics


Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. It is based heavily on combinatorics, particularly the twelvefold way and combinations without replacement.

Choosing 6 from 49

In a typical 6/49 game, each player chooses six distinct numbers from a range of 1-49. If the six numbers on a ticket match the numbers drawn by the lottery, the ticket holder is a jackpot winner—regardless of the order of the numbers. The probability of this happening is 1 in 13,983,816.
The chance of winning can be demonstrated as follows: The first number drawn has a 1 in 49 chance of matching. When the draw comes to the second number, there are now only 48 balls left in the bag, because the balls are drawn without replacement. So there is now a 1 in 48 chance of predicting this number.
Thus for each of the 49 ways of choosing the first number there are 48 different ways of choosing the second. This means that the probability of correctly predicting 2 numbers drawn from 49 in the correct order is calculated as 1 in 49 × 48. On drawing the third number there are only 47 ways of choosing the number; but of course we could have arrived at this point in any of 49 × 48 ways, so the chances of correctly predicting 3 numbers drawn from 49, again in the correct order, is 1 in 49 × 48 × 47. This continues until the sixth number has been drawn, giving the final calculation, 49 × 48 × 47 × 46 × 45 × 44, which can also be written as or 49 factorial divided by 43 factorial. This works out to 10,068,347,520, which is much bigger than the ~14 million stated above.
However; the order of the 6 numbers is not significant. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in. Accordingly, given any set of 6 numbers, there are 6 × 5 × 4 × 3 × 2 × 1 = 6! or 720 orders in which they could be drawn. Dividing 10,068,347,520 by 720 gives 13,983,816, also written as, or more generally as
This function is called the combination function; in Microsoft Excel, this function is implemented as COMBIN. For example, COMBIN, would return 13,983,816. For the rest of this article, we will use the notation. "Combination" means the group of numbers selected, irrespective of the order in which they are drawn.
An alternative method of calculating the odds is to note that the probability of the first ball corresponding to one of the six chosen is 6/49; the probability of the second ball corresponding to one of the remaining five chosen is 5/48; and so on. This yields a final formula of
The range of possible combinations for a given lottery can be referred to as the "number space". "Coverage" is the percentage of a lottery's number space that is in play for a given drawing.

Odds of getting other possibilities in choosing 6 from 49

One must divide the number of combinations producing the given result by the total number of possible combinations. The numerator equates to the number of ways to select the winning numbers multiplied by the number of ways to select the losing numbers.
For a score of n, describes the odds of selecting n winning numbers from the 6 winning numbers. This means that there are 6 - n losing numbers, which are chosen from the 43 losing numbers in ways. The total number of combinations giving that result is, as stated above, the first number multiplied by the second. The expression is therefore.
This can be written in a general form for all lotteries as:
where is the number of balls in lottery, is the number of balls in a single ticket, and is the number of matching balls for a winning ticket.
The generalisation of this formula is called the hypergeometric distribution.
This gives the following results:
ScoreCalculationExact ProbabilityApproximate Decimal ProbabilityApproximate 1/Probability
0435,461/998,8440.4362.2938
168,757/166,4740.4132.4212
244,075/332,9480.1327.5541
38,815/499,4220.017756.66
4645/665,8960.0009691,032.4
543/2,330,6360.000018454,200.8
61/13,983,8160.000000071513,983,816

When a bonus number is included, the adjusted odds are:
ScoreCalculationExact ProbabilityApproximate Decimal ProbabilityApproximate 1/Probability
5, bonus not won0.000018020855,491.33
5, bonus won0.00000042912,330,636

Powerballs and bonus balls

Many lotteries have a Powerball. If the powerball is drawn from a pool of numbers different from the main lottery, the odds are multiplied by the number of powerballs. For example, in the 6 from 49 lottery, given 10 powerball numbers, then the odds of getting a score of 3 and the powerball would be 1 in 56.66 × 10, or 566.6. Another example of such a game is Mega Millions, albeit with different jackpot odds.
Where more than 1 powerball is drawn from a separate pool of balls to the main lottery, the odds of the different possible powerball matching scores are calculated using the method shown in the "other scores" section above, and then multiplied by the odds of achieving the required main-lottery score.
If the powerball is drawn from the same pool of numbers as the main lottery, then, for a given target score, the number of winning combinations includes the powerball. For games based on the Canadian lottery, after the 6 main balls are drawn, an extra ball is drawn from the same pool of balls, and this becomes the powerball. An extra prize is given for matching 5 balls and the bonus ball. As described in the "other scores" section above, the number of ways one can obtain a score of 5 from a single ticket is. Since the number of remaining balls is 43, and the ticket has 1 unmatched number remaining, of these 258 combinations will match the next ball drawn, leaving ways of achieving it. Therefore, the odds of getting a score of 5 and the powerball are.
Of the 258 combinations that match 5 of the main 6 balls, in 42/43 of them the remaining number will not match the powerball, giving odds of for obtaining a score of 5 without matching the powerball.
Using the same principle, the odds of getting a score of 2 and the powerball are for the score of 2 multiplied by the probability of one of the remaining four numbers matching the bonus ball, which is. Since, the probability of obtaining the score of 2 and the bonus ball is, approximate decimal odds of 1 in 81.2.
The general formula for matching balls in a choose lottery with one bonus ball from the pool of balls is:
The general formula for matching balls in a choose lottery with zero bonus ball from the pool of balls is:
The general formula for matching balls in a choose lottery with one bonus ball from a separate pool of balls is:
The general formula for matching balls in a choose lottery with no bonus ball from a separate pool of balls is:

Minimum number of tickets for a match

It is a hard problem to calculate the minimum number of tickets one needs to purchase to guarantee that at least one of these tickets matches at least 2 numbers. In the 5-from-90 lotto, the minimum number of tickets that can guarantee a ticket with at least 2 matches is 100.

Information theoretic results

As a discrete probability space, the probability of any particular lottery outcome is atomic, meaning it is greater than zero. Therefore, the probability of any event is the sum of probabilities of the outcomes of the event. This makes it easy to calculate quantities of interest from information theory. For example, the information content of any event is easy to calculate, by the formula
In particular, the information content of outcome of discrete random variable is
For example, winning in the example above is a Bernoulli-distributed random variable with a chance of winning We write with and. The information content of winning is
shannons or bits of information. The information content of losing is
The information entropy of a lottery probability distribution is also easy to calculate as the expected value of the information content.
Oftentimes the random variable of interest in the lottery is a Bernoulli trial. In this case, the Bernoulli entropy function may be used. Using representing winning the 6-of-49 lottery, the Shannon entropy of 6-of-49 above is