Quoting four posts from the
"Aunt sues nephew over a conjoined lottery ticket" thread:
Quote: (09-02-2018 02:12 PM)Malone Wrote:
...
When you gamble you know the odds, and they're usually not 1:292M.
...
Quote: (09-02-2018 02:27 PM)kmhour Wrote:
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.
...
Quote: (09-02-2018 03:18 PM)porscheguy Wrote:
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>
Quote: (09-03-2018 07:50 AM)Cation Wrote:
Of course the chances of winning the lottery are astronomically low.
Yet, somebody wins from time to time.
_______________________________________________
Several members have mentioned over the years that lotteries have an inherently negative expected value. Here's a technical explanation highlighting how lotteries have so much negative expected value. It also explains how people know, by instinct, to buy lottery tickets when there is a huge jackpot.
I only buy lottery tickets when the expected value of return is positive.
[NOTE: This post shows some math. It will take some time to read.]
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 to find the minimum jackpot to gain a positive return 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 per game/combination,
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 ranging from $200 to more than $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 (5 numbers drawn out of 69 numbers & 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 5 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 per game/combination,
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 approximately 3-5 times a year or never again since lotteries have an inherently negative expected value.
NOTE: The principles of expected value apply to all gambling & investments & is linked to various types of returns.
_______________________________________________
BONUS: Calculating the Mega Millions expected return for October 19, 2018 (11PM ET)
Combinations of Mega Millions (US), a lottery utilising the "extra number":
70
C5 x 25 (5 numbers drawn out of 70 numbers & the 25 are the "yellow ball" segment)
= (70!)/(5!x65!) x 25
= (70x69x68x67x66)/(5x4x3x2x1) x 25
= (7x23x17x67x66) x 25 | [70/(5x2) = 7, 69/3 = 23, 68/4 = 17]
= 12103014 x 25
=
302575350
Therefore, from 70 regular numbers & 25 "yellow balls",
the total amount of 5 number+"yellow ball" combinations is 302 575 350. There are
12 103 014 seven number combinations when the "yellow ball" is not included.
To have positive expected value at $2 per game/combination,
the pre-tax jackpot must be $605 150 700.01 or greater.
2 x 302575350 = 605150700
The estimated
pre-tax expected return is:
(970000000 / 605150700) x 100
=
60.29% positive expected return
In the event of a jackpot win, don't sign the ticket straight away, contact an attorney & always take the annuity. I recommend reading about former MLB player Bobby Bonilla's genius annuity related move on the New York Mets on July 1, 2000.
_______________________________________________
"The worth of a man's life is determined by the life changing events he was fated to experience." #128