Lottery mathematics

Template:Multiple issues Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. It is based primarily on combinatorics, particularly the twelvefold way and combinations without replacement. It can also be used to analyze coincidences that happen in lottery drawings, such as repeated numbers appearing across different draws.^[1] Template:Short description

In the following

Template:Mvar is the number of balls in a pool of balls that the winning balls are drawn from, without replacement.
Template:Mvar is the number of winning balls drawn from the pool.
Template:Mvar is the number of balls listed on the lottery ticket. (It often equals Template:Mvar.)
Template:Mvar is the number of balls that match; these balls are both on the lottery ticket and within the winning set.

Single pool of balls

Suppose there are Template:Mvar unique balls (such as Template:Math) from which balls are to be drawn without replacement. Suppose a subset of Template:Mvar balls (such as Template:Math) is drawn as the winning set. Suppose a subset of Template:Mvar balls (such as Template:Math) is selected on a lottery ticket. Suppose Template:Mvar of the Template:Mvar balls from the lottery ticket are also among the Template:Var balls in the winning set. Out of the $(\binom{P}{W})$ possible ways (see binomial coefficient) to draw the winning set, there are $(\binom{T}{M})$ ways to have Template:Mvar of them come from the Template:Mvar on the lottery ticket and $(\binom{P - T}{W - M})$ ways to have Template:Math of them come from the set of Template:Math not mentioned on the lottery ticket. That is, the probability of getting Template:Mvar matches is given by the following formula when there are Template:Mvar balls in the pool, each lottery ticket selects Template:Mvar balls, and Template:Mvar is the number of winning balls drawn for the lottery. $Pr [M ∣ P, T, W] = \frac{(\binom{T}{M}) (\binom{P - T}{W - M})}{(\binom{P}{W})} .$

This can also be solved from the opposite point of view. There are $(\binom{P}{T})$ ways to fill out the ticket, $(\binom{W}{M})$ ways to choose the matching numbers from winning set, and $(\binom{P - W}{T - M})$ ways to list the remaining numbers in ticket from among the non-winning numbers. $Pr [M ∣ P, T, W] = \frac{(\binom{W}{M}) (\binom{P - W}{T - M})}{(\binom{P}{T})} .$ Both expressions give the same result.

Single pool examples

The chances of getting Template:Mvar matches when drawing Template:Mvar balls from a pool of Template:Mvar balls and lottery tickets with Template:Mvar balls each:

Template:Mvar matches	Template:Math balls from Template:Math	Template:Math balls from Template:Math
0	$\frac{(\binom{6}{0}) (\binom{43}{6})}{(\binom{49}{6})} = \frac{1 \cdot 6,096,454}{13,983,816} \approx 0.436$	$\frac{(\binom{5}{0}) (\binom{64}{5})}{(\binom{69}{5})} = \frac{1 \cdot 7,624,512}{11,238,513} \approx 0.678$
1	$\frac{(\binom{6}{1}) (\binom{43}{5})}{(\binom{49}{6})} = \frac{6 \cdot 962,598}{13,983,816} \approx 0.413$	$\frac{(\binom{5}{1}) (\binom{64}{4})}{(\binom{69}{5})} = \frac{5 \cdot 635,376}{11,238,513} \approx 0.283$
2	$\frac{(\binom{6}{2}) (\binom{43}{4})}{(\binom{49}{6})} = \frac{15 \cdot 123,410}{13,983,816} \approx 0.132$	$\frac{(\binom{5}{2}) (\binom{64}{3})}{(\binom{69}{5})} = \frac{10 \cdot 41,664}{11,238,513} \approx 0.0371$
3	$\frac{(\binom{6}{3}) (\binom{43}{3})}{(\binom{49}{6})} = \frac{20 \cdot 12,341}{13,983,816} \approx 0.0177$	$\frac{(\binom{5}{3}) (\binom{64}{2})}{(\binom{69}{5})} = \frac{10 \cdot 2,016}{11,238,513} \approx 0.001 79$
4	$\frac{(\binom{6}{4}) (\binom{43}{2})}{(\binom{49}{6})} = \frac{15 \cdot 903}{13,983,816} \approx 0.000 969$	$\frac{(\binom{5}{4}) (\binom{64}{1})}{(\binom{69}{5})} = \frac{5 \cdot 64}{11,238,513} \approx 0.000 0285$
5	$\frac{(\binom{6}{5}) (\binom{43}{1})}{(\binom{49}{6})} = \frac{6 \cdot 43}{13,983,816} \approx 0.000 0184$	$\frac{(\binom{5}{5}) (\binom{64}{0})}{(\binom{69}{5})} = \frac{1 \cdot 1}{11,238,513} \approx 0.000 000 0890$
6	$\frac{(\binom{6}{6}) (\binom{43}{0})}{(\binom{49}{6})} = \frac{1 \cdot 1}{13,983,816} \approx 0.000 000 0715$

Power balls from a separate pool of balls

Some lotteries also have one or more power balls drawn from a separate pool of balls. For example, the first drawing may be for Template:Math balls out of Template:Math balls and then Template:Math power ball may be drawn from Template:Math balls. For both pools, the drawing is without replacement. Similarly a lottery ticket will indicate Template:Math regular balls and Template:Math power balls. Often the number of balls on the lottery ticket is the same as the winning set: Template:Math and Template:Math. The probabilities of getting Template:Math matches in the first drawing and Template:Math matches in the power ball drawing is just the product of the individual probabilities: $Pr [M_{1}, M_{2} ∣ P_{1}, T_{1}, W_{1}, P_{2}, T_{2}, W_{2}] = \frac{(\binom{T_{1}}{M_{1}}) (\binom{P_{1} - T_{1}}{W_{1} - M_{1}})}{(\binom{P_{1}}{W_{1}})} \frac{(\binom{T_{2}}{M_{2}}) (\binom{P_{2} - T_{2}}{W_{2} - M_{2}})}{(\binom{P_{2}}{W_{2}})},$ and similarly for three or more pools of balls.

For example, this formula could be used if one ball is drawn from a pool of balls numbered 0 to 9 and then a second and third ball are also drawn from their own pools of 10 balls: Template:Math, and Template:Math. (Equivalently, all three balls could be drawn from the same pool, so long as each ball is returned to the pool before the next ball is drawn.) However, this case is more easily modeled as if a single ball, numbered from 0 (represented as "000") to 999 is drawn from a single pool of 1000 balls.

Separate pool examples

The chances of getting Template:Math matches when drawing Template:Math balls from a pool of Template:Math balls, getting Template:Math matches when drawing Template:Math power balls from a separate pool of Template:Math balls, and having Template:Math regular balls and Template:Math power balls selected on the lottery ticket:

Template:Math matches	Template:Math balls from Template:Math and Template:Math balls from Template:Math
0 + 0	$\frac{(\binom{5}{0}) (\binom{64}{5})}{(\binom{69}{5})} \frac{(\binom{1}{0}) (\binom{25}{1})}{(\binom{26}{1})} = \frac{1 \cdot 7,624,512 \cdot 1 \cdot 25}{11,238,513 \cdot 26} \approx 0.652$
0 + 1	$\frac{(\binom{5}{0}) (\binom{64}{5})}{(\binom{69}{5})} \frac{(\binom{1}{1}) (\binom{25}{0})}{(\binom{26}{1})} = \frac{1 \cdot 7,624,512 \cdot 1 \cdot 1}{11,238,513 \cdot 26} \approx 0.0261$
1 + 0	$\frac{(\binom{5}{1}) (\binom{64}{4})}{(\binom{69}{5})} \frac{(\binom{1}{0}) (\binom{25}{1})}{(\binom{26}{1})} = \frac{5 \cdot 635,376 \cdot 1 \cdot 25}{11,238,513 \cdot 26} \approx 0.272$
1 + 1	$\frac{(\binom{5}{1}) (\binom{64}{4})}{(\binom{69}{5})} \frac{(\binom{1}{1}) (\binom{25}{0})}{(\binom{26}{1})} = \frac{5 \cdot 635,376 \cdot 1 \cdot 1}{11,238,513 \cdot 26} \approx 0.001 09$
2 + 0	$\frac{(\binom{5}{2}) (\binom{64}{3})}{(\binom{69}{5})} \frac{(\binom{1}{0}) (\binom{25}{1})}{(\binom{26}{1})} = \frac{10 \cdot 41,664 \cdot 1 \cdot 25}{11,238,513 \cdot 26} \approx 0.0356$
2 + 1	$\frac{(\binom{5}{2}) (\binom{64}{3})}{(\binom{69}{5})} \frac{(\binom{1}{1}) (\binom{25}{0})}{(\binom{26}{1})} = \frac{10 \cdot 41,664 \cdot 1 \cdot 1}{11,238,513 \cdot 26} \approx 0.001 43$
3 + 0	$\frac{(\binom{5}{3}) (\binom{64}{2})}{(\binom{69}{5})} \frac{(\binom{1}{0}) (\binom{25}{1})}{(\binom{26}{1})} = \frac{10 \cdot 2,016 \cdot 1 \cdot 25}{11,238,513 \cdot 26} \approx 0.001 72$
3 + 1	$\frac{(\binom{5}{3}) (\binom{64}{2})}{(\binom{69}{5})} \frac{(\binom{1}{1}) (\binom{25}{0})}{(\binom{26}{1})} = \frac{10 \cdot 2,016 \cdot 1 \cdot 1}{11,238,513 \cdot 26} \approx 0.000 0690$
4 + 0	$\frac{(\binom{5}{4}) (\binom{64}{1})}{(\binom{69}{5})} \frac{(\binom{1}{0}) (\binom{25}{1})}{(\binom{26}{1})} = \frac{5 \cdot 64 \cdot 1 \cdot 25}{11,238,513 \cdot 26} \approx 0.000 0274$
4 + 1	$\frac{(\binom{5}{4}) (\binom{64}{1})}{(\binom{69}{5})} \frac{(\binom{1}{1}) (\binom{25}{0})}{(\binom{26}{1})} = \frac{5 \cdot 64 \cdot 1 \cdot 1}{11,238,513 \cdot 26} \approx 0.000 001 10$
5 + 0	$\frac{(\binom{5}{5}) (\binom{64}{0})}{(\binom{69}{5})} \frac{(\binom{1}{0}) (\binom{25}{1})}{(\binom{26}{1})} = \frac{1 \cdot 1 \cdot 1 \cdot 25}{11,238,513 \cdot 26} \approx 0.000 000 0856$
5 + 1	$\frac{(\binom{5}{5}) (\binom{64}{0})}{(\binom{69}{5})} \frac{(\binom{1}{1}) (\binom{25}{0})}{(\binom{26}{1})} = \frac{1 \cdot 1 \cdot 1 \cdot 1}{11,238,513 \cdot 26} \approx 0.000 000 003 42$

Bonus balls from the same pool of balls

Some lotteries have one or more bonus balls drawn from the original pool of balls after the first round of balls is drawn. In this scenario each lottery ticket indicates Template:Mvar balls out of Template:Mvar possibilities, but the drawing is for Template:Math balls plus Template:Math bonus balls, without replacement. The probability that Template:Math balls from the first drawing match the lottery ticket and Template:Math balls from the bonus-ball drawing match the lottery ticket is given by $Pr [M_{1}, M_{2} ∣ P, T, W_{1}, W_{2}] = \frac{(\binom{W_{1}}{M_{1}}) (\binom{W_{2}}{M_{2}}) (\binom{P - W_{1} - W_{2}}{T - M_{1} - M_{2}})}{(\binom{P}{T})} .$

Example of balls and bonus balls from the same pool

When Template:Math of Template:Math numbers are on a lottery ticket but the winning set is Template:Math numbers plus Template:Math bonus ball then probabilities are as follows.

Template:Math matches	Template:Math balls on ticket, Template:Math balls and Template:Math bonus ball drawn from Template:Math balls
0 + 0	$\frac{(\binom{6}{0}) (\binom{1}{0}) (\binom{42}{6})}{(\binom{49}{6})} = \frac{1 \cdot 1 \cdot 5,245,786}{13,983,816} \approx 0.375$
0 + 1	$\frac{(\binom{6}{0}) (\binom{1}{1}) (\binom{42}{5})}{(\binom{49}{6})} = \frac{1 \cdot 1 \cdot 850,668}{13,983,816} \approx 0.0608$
1 + 0	$\frac{(\binom{6}{1}) (\binom{1}{0}) (\binom{42}{5})}{(\binom{49}{6})} = \frac{6 \cdot 1 \cdot 850,668}{13,983,816} \approx 0.365$
1 + 1	$\frac{(\binom{6}{1}) (\binom{1}{1}) (\binom{42}{4})}{(\binom{49}{6})} = \frac{6 \cdot 1 \cdot 111,930}{13,983,816} \approx 0.0480$
2 + 0	$\frac{(\binom{6}{2}) (\binom{1}{0}) (\binom{42}{4})}{(\binom{49}{6})} = \frac{15 \cdot 1 \cdot 111,930}{13,983,816} \approx 0.120$
2 + 1	$\frac{(\binom{6}{2}) (\binom{1}{1}) (\binom{42}{3})}{(\binom{49}{6})} = \frac{15 \cdot 1 \cdot 11,480}{13,983,816} \approx 0.0123$
3 + 0	$\frac{(\binom{6}{3}) (\binom{1}{0}) (\binom{42}{3})}{(\binom{49}{6})} = \frac{20 \cdot 1 \cdot 11,480}{13,983,816} \approx 0.0164$
3 + 1	$\frac{(\binom{6}{3}) (\binom{1}{1}) (\binom{42}{2})}{(\binom{49}{6})} = \frac{20 \cdot 1 \cdot 861}{13,983,816} \approx 0.00123$
4 + 0	$\frac{(\binom{6}{4}) (\binom{1}{0}) (\binom{42}{2})}{(\binom{49}{6})} = \frac{15 \cdot 1 \cdot 861}{13,983,816} \approx 0.000 924$
4 + 1	$\frac{(\binom{6}{4}) (\binom{1}{1}) (\binom{42}{1})}{(\binom{49}{6})} = \frac{15 \cdot 1 \cdot 42}{13,983,816} \approx 0.000 0451$
5 + 0	$\frac{(\binom{6}{5}) (\binom{1}{0}) (\binom{42}{1})}{(\binom{49}{6})} = \frac{6 \cdot 1 \cdot 42}{13,983,816} \approx 0.000 0180$
5 + 1	$\frac{(\binom{6}{5}) (\binom{1}{1}) (\binom{42}{0})}{(\binom{49}{6})} = \frac{6 \cdot 1 \cdot 1}{13,983,816} \approx 0.000 000 429$
6 + 0	$\frac{(\binom{6}{6}) (\binom{1}{0}) (\binom{42}{0})}{(\binom{49}{6})} = \frac{1 \cdot 1 \cdot 1}{13,983,816} \approx 0.000 000 0715$
6 + 1	Impossible

Ensuring to win the jackpot

There is only one way to ensure winning the jackpot; it is to buy at least one lottery ticket for every possible number combination. For example, one has to buy 13,983,816 different tickets to ensure winning the jackpot in a 6/49 game. To be profitable, the cost of acquiring these tickets (including any overhead) must not exceed the total amount that those tickets will win, including jackpots and any smaller prizes. If it is likely that the jackpot (or any of the smaller prizes) will have to be split among several winners then only your likely share should be counted in the total amount that those tickets will win.

Minimum number of tickets for a match

It is a hard (and often open) 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.^[2]

Coincidences involving lottery numbers

Coincidences in lottery drawings often capture our imagination and can make news headlines as they seemingly highlight patterns in what should be entirely random outcomes. For example, repeated numbers appearing across different draws may appear on the surface to be too implausible to be by pure chance. For instance, on September 6, 2009, the six numbers 4, 15, 23, 24, 35, and 42 were drawn from 49 in the Bulgarian national 6/49 lottery, and in the very next drawing on September 10th, the same six numbers were drawn again. Lottery mathematics can be used to analyze these extraordinary events.^[1]

References

Template:Reflist

External links

Euler's Analysis of the Genoese Lottery – Convergence (August 2010), Mathematical Association of America
Lottery Mathematics – INFAROM Publishing
13,983,816 and the Lottery – YouTube video with James Clewett, Numberphile, March 2012

↑ ^1.0 ^1.1 M. Pollanen (2024) A Double Birthday Paradox in the Study of Coincidences, Mathematics 23(24), 3882. https://doi.org/10.3390/math12243882
↑ Template:Cite journal [1]

[Pollanen-1] 1.0 ^1.1 M. Pollanen (2024) A Double Birthday Paradox in the Study of Coincidences, Mathematics 23(24), 3882. https://doi.org/10.3390/math12243882

[2] Template:Cite journal [1]

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Lottery mathematics

Contents

Single pool of balls

Single pool examples

Power balls from a separate pool of balls

Separate pool examples

Bonus balls from the same pool of balls

Example of balls and bonus balls from the same pool

Ensuring to win the jackpot

Minimum number of tickets for a match

Coincidences involving lottery numbers

References

External links

Navigation menu

Lottery mathematics

Single pool of balls

Single pool examples

Power balls from a separate pool of balls

Separate pool examples

Bonus balls from the same pool of balls

Example of balls and bonus balls from the same pool

Ensuring to win the jackpot

Minimum number of tickets for a match

Coincidences involving lottery numbers

References

External links

Navigation menu

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