# Why you should not play the lottery

Posted by Marcus - 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 = 1000000year_all = np.arange(years)win_chance = np.empty(years)win_chance = 100*(1-(chance**(year_all*52))) Plotting this result in the figure seen below. 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): near_search = np.abs(array-value) idx = (near_search).argmin() return idxfind_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.