Lottery mathematics is used here to mean the calculation of the
probabilities in a
lottery
game. The lottery game used in the examples below is one in which one
selects 6 numbers from 49, and hopes that as many of those 6 as possible
match the 6 that are randomly selected from the same pool of 49 numbers
in the "draw".
Calculation explained in choosing 6 from 49
In a typical 6/49 game, six numbers are drawn from a range of 49 and
if the six numbers on a ticket match the numbers drawn, the ticket
holder is a
jackpot winner—this is true no matter in which order the numbers appear. The probability of this happening is 1 in 13,983,816.
This small
chance of winning can be demonstrated as follows:
Starting with a bag of 49 differently-numbered lottery balls, there
are 49 different but equally likely ways of choosing the number of the
first ball selected from the bag, and so there is a 1 in 49 chance of
predicting
the number correctly. When the draw comes to the second number, there
are now only 48 balls left in the bag (because the balls already drawn
are not returned to the bag) 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 gotten to
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
. This works out to a very large number, 10,068,347,520, which is much bigger than the 14 million stated above.
The last step is to understand that 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 49! / (6! × (49 - 6)!), or more
generally as
- .
This function is called the
combination function; in
Microsoft Excel, this function is implemented as COMBIN(
n,
k).
For example, COMBIN(49, 6) (the calculation shown above), 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 never make the
erroneous assumption that balls must be selected in a certain order. The
odds of the first ball corresponding to one of the six chosen is 6/49;
the odds 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 (for example,
,
as explained in the section above). The numerator equates to the number
of ways one can select the winning numbers multiplied by the number of
ways one can select the losing numbers.
For a score of
n (for example, if 3 of your numbers match the 6 balls drawn, then
n = 3), there are
ways 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 (the HYPGEOMDIST() function in most popular spreadsheets).
This gives the following results:
Score |
Calculation |
Exact Probability |
Approximate Decimal Probability |
Approximate 1/Probability |
0 |
|
435,461/998,844 |
0.436 |
2.2938 |
1 |
|
68,757/166,474 |
0.413 |
2.4212 |
2 |
|
44,075/332,948 |
0.132 |
7.5541 |
3 |
|
8,815/499,422 |
0.0177 |
56.66 |
4 |
|
645/665,896 |
0.000969 |
1,032.4 |
5 |
|
43/2,330,636 |
0.0000184 |
54,200.8 |
6 |
|
1/13,983,816 |
0.0000000715 |
13,983,816 |
Powerballs And Bonus Balls
Many lotteries have a
powerball (or "bonus ball"). If the powerball is drawn from a pool of numbers
different
from the main lottery, then simply multiply the odds by the number of
powerballs. For example, in the 6 from 49 lottery, if there were 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 (the
probability would be divided by 10, to give an exact value of 8815/4994220). 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 (for example, in the
Euromillions game), the odds of the different possible powerball matching scores should be calculated using the method shown in the "
other scores"
section above (in other words, treat the powerballs like a mini-lottery
in their own right), 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, one must calculate the
number of winning combinations, including the powerball. For games based
on the
Canadian lottery (such as the
United Kingdom's
lottery), after the 6 main balls are drawn, an extra ball is drawn from
the same pool of balls, and this becomes the powerball (or "bonus
ball"), and there is an extra prize 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
or 258. Since the number of remaining balls is 43, and the ticket has 1
unmatched number remaining, 1/43 of these 258 combinations will match
the next ball drawn (the powerball). So, there are 258/43 = 6 ways of
achieving it. Therefore, the odds of getting a score of 5 and the
powerball are
= 1 in 2,330,636.
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
= 3/166,474 (approximately 55,491.33) for obtaining a score of 5 without matching the powerball.
Using the same principle, to calculate the odds of getting a score of
2 and the powerball, calculate the number of ways to get a score of 2
as
= 1,851,150 then multiply this by the probability of one of the
remaining four numbers matching the bonus ball, which is 4/43. Since
1,851,150 × (4/43) = 172,200, the probability of obtaining the score of 2
and the bonus ball is
= 1025/83237. This gives approximate decimal odds of 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, in most cases open, mathematical 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 that can guarantee a ticket with at least 2
matches is 100.
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