Calculating Lottery Odds & Their Expected Value (Datasheet)
Malone said:
...
When you gamble you know the odds, and they're usually not 1:292M.
...
kmhour said:
You could less aggressively refer to the lottery as a tax on people who can't do math.
Not that I haven't bought a ticket a few times when the jackpot has been astronomically high. Or participated in a few work pools.
...
porscheguy said:
I’ll buy when the jackpot gets absurdly high. Even though your chance at winning is essentially zero, I’ll still drop $20 on some tickets. Because my chance is the same as anyone else. And it’s not like I’m skipping meals to do it.
<snip>
Cation said:
Of course the chances of winning the lottery are astronomically low.
Yet, somebody wins from time to time.
[NOTE: This post shows some math. It'll take some time to read. If there's enough interest, I may do the mathematics of loans in a future post.]
I only buy lottery tickets when the expected value of return is zero or positive.
The formula for expected return in lotteries is
To calculate the required jackpot for zero or positive expected value, use the below formula:
, then rearrange to get
Therefore, the formula is
These are maths of a majority of lottery draws (combinations):
Where:
n represents the amount of numbers in the barrel &
r represents the amount of numbers drawn from the barrel
Example of a lottery using 45 numbers & drawing 7 numbers:
45
C7
= (45!)/(7!x38!)
= (45x44x43x42x41x40x39)/(7x6x5x4x3x2x1)
= 3x44x43x41x5x39 [42/(7x6) cancels out, 45/(5x3) = 3 & 40/(4x2) = 5]
= 45379620
Therefore, in a 45 number lottery,
the total amount of 7 number combinations is 45 379 620.
To have positive expected value at, let's say,
$1.30 (one game),
the jackpot must be $58 993 504.71 or greater.
1.3 x 45379619 = 58993504.7
Lottery jackpots only become massive when the amount is significantly higher than the total number of combinations. From the previous example, a >$90 million jackpot is massive. Lotteries with jackpots reaching $200 to >$500 million would probably have it set up to have over 100s of millions of drawn number combinations.
Combinations of Powerball (US), a lottery utilising the "extra number":
69
C5 x 26 (the 26 are the "extra number" segment)
= (69!)/(5!x64!) x 26
= (69x68x67x66x65)/(5x4x3x2x1) x 26
= (69x17x67x11x13) x 26 [68/4 = 17, 66/(3x2) = 11, 65/5 = 13]
= 11238513 x 26
= 292201338
Therefore, from 69 regular numbers & 26 "Powerballs",
the total amount of 7 number+Powerball combinations is 292 201 338. There are
11 238 513 seven number combinations when the "Powerball" is not included.
When calculating the lottery odds, calculate the combination component first, then multiply that number of combinations by the "extra number" if the lottery uses one.
To have positive expected value at, let's say,
$2 (one game),
the pre-tax jackpot must be $584 402 674.01 or greater.
2 x 292201337 = 584402674
If you follow the maths of your local lotteries by substituting the relevant numbers in the formula you'll know when to buy a ticket, which would either be 3-5 times a year or never again since lotteries have an inherent negative expected value.
PS: The principles of expected value apply to all gambling & investments & is linked to various types of returns.