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Mathematics of
Lotto
Lotteries have the worst odds
of any form of gambling,
with a chance of approximately 1 in 14 million for the big win when
playing most lottery jackpot games. However, they promise the greatest
potential payoff to the winner with prizes regularly amounting to
millions of dollars.
Scratch-Off Tickets
In instant scratch-off games, the state takes about 50% of the action.
34% going to public education and the rest covering administrative and
promotional costs. The remaining 50% is returned in prizes. Since the
state keeps 50 cents of every dollar bet, the house edge is 50%, making
the scratch-off game worse than any game in Nevada.
To find the probability of winning a
particular prize, take the number of tickets for that particular prize
and divide by the total number of tickets.
"Probability of
Winning" = "No. of Winning Tickets" / "Total Amount
of Tickets"
Example: Win and Spin.
Win and Spin was a typical California Lottery instant scratch-off game.
There were 135 million tickets printed for Win and Spin, with 22,768,623
winners. This made the chance of winning equal 22,768,623/135,000,000,
or about 1 in 6. Here are the Win and Spin payoffs, number of tickets of
each type, and probabilities.
| Instant
Prize (in $) |
No.
of Winners |
Probability |
or |
| 1 |
10,800,000 |
0.083 |
1
in 12 |
| 2 |
8,100,000 |
0.062 |
1
in 16 |
| 5 |
3,240,000 |
0.025 |
1
in 40 |
| 10 |
540,000 |
0.004 |
1
in 241 |
| 50 |
54,000 |
4.153-4 |
1
in 2,407 |
| 100 |
27,000 |
2.076-4 |
1
in 4,815 |
| 500 |
6,073 |
4.672-5 |
1
in 21,406 |
| 1,000 |
1,350 |
1.038-5 |
1
in 96,296 |
| 10,000 |
150 |
1.154-6 |
1
in 866,667 |
| Big
Spin |
50 |
3.846-7 |
1
in 2,600,000 |
To find your expected payoff, multiply each payoff by
the number of winners, add, and divide the resulting sum by the total
amount of tickets.
"Expected
Payoff" = [Sum ("Payoff"
x "No. of Winning Tickets)] /
"Total Amount of Tickets"
where [Sum ("Payoff"
x "No. of Winning Tickets)] simply is
the total amount of prizes given away.
Example: Win and Spin.
[(1 x 10,800,000)
+ (2 x 8,100,000)
+ (5 x 3,240,000)
+ (10 x 540,000)
+ (50 x 54,000)
+ (100 x 27,000)
+ (500 x 6,073)
+ (1,000 x 1,350)
+ (10,000 x 150)]
/ 130,000,000
=
[10,800,000 + 16,200,000 + 16,200,000 + 5,400,000 + 2,700,000 + 2,700,000
+ 3,036,500 + 1,350,000 + 1,500,000] / 130,000,000
=
59,886,500 / 130,000,000
=
0.46066538 or ca. 0.46
The result is an expected payoff of about
+$.46. Now you have to subtract the prize for the ticket. Subtracting
let's say one dollar you paid for the ticket leaves you with expected
winnings = -$.54.
Lotto
In order to develop a formula for computing
lotto probabilities, we must get familiar with the world of
combinatorics (counting).
Example: Lotto: Pick 6 out of 49
numbers.
The question is: What is the chance that
6 of your 6 picks are amongst the 6 winning numbers?
The next question is: What is the chance that 5 of your 6 picks are
amongst the 6 winning numbers?
And so on.
To start with we first need to know in
how many ways 49 numbers can be drawn. The first ball could be any of 49
available balls. The second ball could be any of 48 available balls (because one is drawn
already). The third ball could be any of the 47
remaining balls. The next ball could be any of 46 remaining balls. And
so on. All the way down to the last of the 49 balls. Thus there are 49 x
48 x 47x 46 x 45 x 44 x 43 x 42 x 41 x ... x 1 = 6.0828262 ways
in which 49 balls can be drawn. Mathematically you say there are
"49 factorial" ways and write "49!".
Now we need to know in how many ways you
can draw 6 out of 49 numbers. We don't ask for the chance to draw 6
winning numbers yet. Could be any number. Well, there are n! / k! (n_k)!
ways to select k numbers from n numbers. This formula is read as "n
choose k" and for convenience denoted by (nk).
In our example there are (496) = 49! / 6! (49-6)!
= 13,983,816 or nearly 14 million ways to select 6 out of 49 numbers.
So the chance of winning the jackpot is 1 in 14 million. To put these
odds in perspective, if you buy 50 lotto tickets a week, you'll win the
jackpot about once every 5,000 years. Or suppose you are in a football
stadium filled with 70,000 people, and there are 200 such stadiums.
Select one person at random from these 200 stadiums. Your chance of
being selected is the same as your chance of winning the Lotto jackpot.
Now we want to know what the probability
is that only 1of 6 select numbers is a winning number:
a) There are
(496) = 13,983,816
ways to select 6 out of 49 numbers.
b) There are
6 winning numbers. So there are (61) = 6! / 1!
(6-1)! = 6 ways to select 1 out of 6
winning numbers.
c) There are
49 - 6 = 49! / 6! (49-6)! = 43 numbers that
don't win. So there are (435) = 43! / 5! (43-5)! =
962,596 ways to select 5 out of 43 numbers that don't win.
d) This
gives a total of (61)
x (435) = 5,775,588
ways to select 6 out of 49 numbers, with one that is a winning number.
To get the probability of having one
number correct you just need to divide "the
number of ways to select 6 out of 49 numbers, with one that is a winning
number" by "the total amount of
ways to select 6 out of 49 numbers". Thus,
| P(0
winning numbers) = (60)
x (436) /
(496) = |
6,096,454
/ 13,983,816
= |
0.43596497 |
or 1 in 2 |
|
P(1
winning number) = (61)
x (435) /
(496) = |
5,775,588
/ 13,983,816
= |
0.41301945 |
or 1 in 2 |
| P(2
winning numbers) = (62)
x (434) /
(496) = |
1,851,150
/ 13,983,816
= |
0.13237802 |
or 1 in 8 |
| P(3
winning numbers) = (63)
x (433) /
(496) = |
246,820
/ 13,983,816
= |
0.01765040 |
or 1 in 57 |
| P(4
winning numbers) = (64)
x (432) /
(496) = |
13,545
/ 13,983,816
= |
0.00096819 |
or 1 in
1,032 |
| P(5
winning numbers) = (65)
x (431) /
(496) = |
258
/ 13,983,816
= |
0.00001844 |
or 1 in
54,201 |
| P(6
winning numbers) = (66)
x (430) /
(496) = |
1
/ 13,983,816
= |
0.00000007 |
or 1 in
13,983,816 |
To compute the expected payoff for a
lottery ticket, multiply the payoffs for each number_match by their
probabilities to occur and add them all together. Subtract from this
amount the price for the ticket.
Since the payoffs are depending on a) how much money the lottery pot
contains, and b) how many people have won, and splitting prize money with
the winners means that there are no fixed payoff odds, it is not possible
to calculate the expected payoff for each lottery ticket you buy.
As with instant scratch-off games, 50% of
the money bet on Lotto is kept by the state and 50% is returned in prizes.
In a sense, the house edge is 50%.
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