Why you should not play the lottery

Lotteries are bad, m'kay

Published on December 11, 2015

In Denmark, chance gambling is popular, very popular. In 2014 Danske Spil had net earnings of 1560 M dkk, which is roughly 229 M \$.
A significant part of this stems from Onsdags Lotto (Wednesday lottery).

Onsdags Lotto, the numbers

To win the grand prize in Onsdags Lotto you need to hit all $r=6$ correct numbers on your ticket. These numbers are extracted using $n=48$ numbered balls. When the first number is drawn we thus have 48 possibilities, when the second number is drawn we have 47 possibilities and so on, until all 6 numbers have been drawn. We can thus calculate the chance of winning Onsdags lotto on a single ticket to be

$$_{n, r}C = \frac{n!}{(n-r)! \cdot r!} = 12271512$$

i.e. roughly 1 in 12 M.
We can now write a more general expression for the chance of winning starting with the probability of losing when taking into account the number of tickets per week, $g$, and number of weeks played, $w$. 

$$_{n, r}^{g, w}C_{lose} = (1-(_{n, r}C)^{-g})^{w}$$

The final expression for winning (in percentages) then looks like

$$_{n, r}^{g, w}C_{\%} = 100 \cdot (1-_{n, r}^{g, w}C_{lose})$$

Let's play

We start by buying a single lottery ticket in a weeks game. We can then calculate our winning chance to be 

$$_{48, 6}^{1, 1}C_{\%} = 8.15 \cdot 10^{-6} \%$$

That is really not alot.
We now do this every week for 50 years where we use 52 weeks in a year

$$_{48, 6}^{1, 50 \cdot 52}C_{\%} = 0.021 \%$$

Again, not really good odds yet.
What if we happened to be immortal and did this for 1 M years?

$$_{48, 6}^{1, 1000000 \cdot 52}C_{\%} = 98.56 \%$$

That is more like it!

Accumulated probability

Using the equations presented above we can calculate the probability for all years from 0 to 1 M. In Python this can done like so

years = 1000000<br />year_all = np.arange(years)<br />win_chance = np.empty(years)<br /><br />win_chance = 100*(1-(chance**(year_all*52)))

Plotting this result in the figure seen below.

lottery

Years played on the x-axis and accumulated chance of winning on the y-axis.

Playing until 25 % chance 

A million years is a long time though, what if we just settled for getting 25 % chance of winning. How many years should we play to reach that?
We can extract this information from the plot above using this command in Python

def find_nearest_ind(array, value):<br />    near_search = np.abs(array-value)<br />    idx = (near_search).argmin()<br />    return idx<br /><br />find_nearest_ind(win_chance,25))

This returns a value of 67890 years to reach 25% chance of winning. Thats a lot less than 1 M years from before but still too much. If we instead buy 10000 lottery tickets per week we will be hitting 25% after just 7 years!

However, playing the lottery is not free so we have to take the expenses into account as well. A single Onsdags Lotto ticket cost 50 dkk or around 7 \$ and the typical grand prize is around 5 M dkk or roughly 0.73 M \$. We can thus write our net earnings as

$Net = Prize - Tickets$
$\ \ \ \ \ \ \ = 0.73 M \ \$ - 7 y \cdot 70000 \frac{\$}{week}$ 
$\ \ \ \ \ \ \ \approx -24.8 M \$$

I.e. we lost 24.8 M \$ or roughly 169 M dkk.

This is the reason why you should not play the lottery.

The full code for this post can be found at Github.