According to rational (or statistical) analysis
Photo by Alain Pham on UnsplashIs there ever a time the expected return would be positive from a lottery ticket?
More people tend to buy lottery tickets when there is a larger jackpot. Does this make sense?
This article aims to answer these questions based on statistical analysis.
Introduction
The lottery we are studying here is one of the most popular lottery games in Canada — Lotto 6/49. Like many other lottery games, Lotto 6/49 also rolls over the jackpot when there are no winning tickets in a draw. Over the past year, there have been a few large jackpots of over $20 million.
Jackpot size for Lotto 6/49 over the past yearThe largest single jackpotever in Canadian lottery history is from Lotto 6/49 as well. It was drawn on October 17, 2015 for a jackpot of $64 million. And the jackpot was won by one single ticket purchased in Ontario. This is less than 10% of the American largest jackpot in history — $768.4 million jackpot in the Powerball. But it corresponds to the Canadian population vs U.S. population ratio.
There are 7 different prizes for the main draw according to number matches. There is also a $1 million guaranteed prize draw which goes to one of the tickets in play. The odds of winning any prize (most likely, another free play) is quite good at one in 6.6.
However, no one is targeting small prizes when buying lottery tickets. The main jackpot is what we are mainly interested in.
The odds of winning the Lotto 6/49 main jackpot is 1 in 13,983,816! That means, we have a higher chance of:
- being struck by lightning — 1 in 114,195.
- becoming a movie star — 1 in 1,190,000.
- getting attacked by a shark — 1 in 3,748,067.
Things to consider before the analysis
- Taxes and time value of money
Luckily, the lottery winner in Canada does not have to pay taxes on the winnings — only on the income it generates. The winners seem to have the option of getting paid in one lump sum after claiming. Therefore, there is no need to consider the time value of the winning amount and the taxes, which might reduce the actual prize.
Unlike in Powerball, the Lotto 6/49 jackpot amount is simply the amount the winner gets!
- Data
Due to data limitations, only one year of Lotto 6/49 history was available for analysis¹. Other data is read from news reports.
- Model used
Lastly, we also assume the number of jackpot winners follows a Poisson distribution. A Poisson distribution is a probability distribution often used for modeling the number of rare events observed in some specific interval, such as time.
This is not a perfect model but it’s simple and can be used as a good proxy. Details about the application can be found here.
The math
When are the best times to buy a lottery ticket?
To answer this question, we use expected return as the standard measure.
*******You may skip this part if you’re not interested in the calculations*******
We mentioned the chance of winning the jackpot with one ticket is one in 13,983,816. Each ticket is considered independent from each other, so we’ll assume this to be constant.
The expected return of the main prize jackpot is calculated as:
(expected amount of winnings given that you win) * (1/13983816).
The expected amount of winnings given that you win is derived from the Poisson distribution assumption we made.
The expected return of prizes from the main draw other than the main jackpot was simply calculated as:
(the amount of winnings)* (odds of winning for that particular prize).
The expected return of the guaranteed $1 million prize is calculated as:
1 million * (1/total number of tickets sold).²
******************************************************************
The purchase price per ticket is $3. Hence, the total expected return = sum (expected returns from different prizes) - $3.
Below is a chart with the data of expected total return and the jackpot size over the past year.
Plot of jackpot size (x-axis) and expected total return (y-axis)The relationship looks linear. The expected return of a jackpot of $33 million (the largest jackpot in the past year) was as high as -$0.14.
The higher the jackpot, the better the expected return?
Does that mean it will reach 0 or even positive with a higher jackpot, which will make the lottery a good bet?
To answer this question, let’s look at a bigger jackpot that happened back in 2005. The jackpot was $54.3 million, which is a lot bigger than $33 million and closer to the highest record of $64 million. This particular draw was picked due to data availability from a news report.
Back then the tickets were $2 each so it was considered a huge jackpot equivalent to an $81.45 million jackpot today. Even so, the expected return³ was -$0.14, which is the same as the highest expected value of the past year.
Why is that?
Because there were almost 50 million tickets sold for this jackpot. This is more than ten times compared to the average draw (4.9 million) in the past year. People are definitely excited and buy more tickets for these larger jackpots! And that is the main factor that pulls down the expected return.
Yes, the expected return of a lottery ticket does increase as the jackpot gets higher.
When the jackpot gets large, it is a better time to buy the lottery.
But it will never become a rational investment (with positive expected return) due to the lotto mania created by these large jackpots!
Moreover, the more tickets sold, the more likely the winner will have to split the prize with others. This will largely decrease the value even if you do win the jackpot. So let’s move onto the other question.
How likely would you have to split the prize with others when the jackpot is getting larger?
- Jackpot of a medium size
First, take a look at the $33 million jackpot — the largest one in the past year. The number of tickets sold was estimated to be 7.6 million.
Given that we can’t change the odds of winning the jackpot. The concern is if I do win, there could be multiple winners that I have to split the winnings with.
Based on the above information, we are able to calculate the probability of different scenarios, given you won the lottery.
The scenarios and distribution given you already won $33 millionThis is looking very good. There’s a large chance (58%) you are the only winner and the prize will be the full size of $33 million.
But as mentioned earlier, as the jackpot gets bigger, the number of tickets sold will grow at a non-linear relationship.
- Jackpot of historically large size
Let’s use the jackpot of $54.3 million that happened back in 2005 as an example again.
The scenarios and distribution given you already won $53.4 millionNow the story is completely different. This chart shows that most likely you’ll have to split the prize between 2–4 other winners, which would largely reduce the amount you can claim, even if you do win the jackpot.
The larger the jackpot, the more likely you will have to split the winning amount with other winners.
The popular draws will give you very small chance of being able to take the full prize home alone!
Summary
The above analysis is done on Canadian Lotto 6/49. However, similar ideas/results should apply to other lottery games such as Powerball in the US.
In short, the lottery is never a rational investment. The best time to buy is when the jackpot is very big!
Note from Towards Data Science’s editors: While we allow independent authors to publish articles in accordance with our rules and guidelines, we do not endorse each author’s contribution. You should not rely on an author’s works without seeking professional advice. See our Reader Terms for details.
¹ The data was scraped from the official website of Lotto 6/49. Only one year of data is currently available.
² Due to data limitations, we used the formula described on the official website of Lotto 6/49 to estimate total tickets sold based on the number of winners and prize amounts.
³ Due to data limitations, the expected value of the prizes other than the main jackpot was calculated as the average of the past year’s draws.