Why Betting on the Lottery is a Pretty Bad Idea (if you actually wish to win)

Why Betting on the Lottery is a Pretty Bad Idea (if you actually wish to win)

Using data visualisation to put into perspective the extent to which we fail at gauging the outcomes of our decisions

We’ve heard time and time again that it’s hard to win the lottery. But yet we still play it.

Doesn’t a chance of 1 in 13,983,816* to hit the jackpot sound close to impossible and daunting? Apparently not.

Or that 435,461/998,844* of the time you end up with 0 matching numbers and don’t earn anything? Not discouraging enough, it seems.

Why then do people fail to get a feel for how remote of a chance winning the jackpot really is? Is it due to our inability to comprehend the sheer magnitude of these huge numbers? To figure that out, I decided to find a way to concretise the true meaning behind these numbers.

*Probabilities derived from a standard 6 from 49 (6/49) lottery. If you’re thinking about other lottery variations like Powerball or Mega Millions, the winning odds are much worse.

The Visualisation

Just yesterday I thought of an idea to visualise all the possible outcomes of choosing 6 numbers (or balls) from 49 numbers.

I would then group these 13,983,816 different outcomes into 7 different sets (or events), with each set representing all the 6-number combinations that would have an exact match of k (where k is any integer between 0 and 6) numbers to an arbitrary winning 6-number combination.

If you are getting REALLY confused with the terms outcomes and events by now, here’s a quick 1-minute crash course:

Trial: a procedure that generates an outcome (i.e. the act of choosing 6 numbers out of 49)

Outcome: a possible result of a trial (i.e. the numbers that you have drawn, [8, 13, 34, 35, 41, 49] )

Event: a set containing outcomes of an experiment that fulfil a certain criteria (i.e. the set of outcomes that have 6 exact matches with the winning combination [1, 2, 3, 4, 5, 6] is the set containing the sole outcome [1, 2, 3, 4, 5, 6])

Each outcome would be represented by a pixel, and all the 13,983,816 pixels representing all the outcomes would be plotted side-by-side to give an indication of how large each set is with respect to one another.

Lo and behold, the visualisation is not for the faint-hearted:

Outcomes of drawing 6-number combination from a set of 49 numbers that matches with a winning 6-number combination

Now before you actually start trying to read the axes like you would do on a normal graph, note that the numbers on the side and the bottom do NOT form part of any axes. They are simply quantity markers to help you count the pixels (which represent outcomes) in the graphic, especially since there are so many of them.

So what does this mean? The set of having 0 matching numbers fills up close to almost half of the entire outcome space, indicating that the probability of getting 0 numbers right is well, close to half.

We then see the coloured blocks getting smaller and smaller. At 3 matching numbers (the orange block), things start to get tiny and I added 2 magnified portions to help you appreciate their relative sizes.

But wait, what’s with that tiny little red dot at the top left corner? Yes, it is what you think it is, a pixel representing the sole outcome of hitting the jackpot. And it even has to be magnified for it to be seen.

Not convinced that it’s small? Try using the quantity markers to help you see how minuscule a single pixel is compared the entire plot — the length of the plot is about ~3500 pixels long. Same goes for its height.

Still think that it’s easy to select the winning jackpot combination?

Seeing Things Differently— The Urn and the Ball

If you find it difficult to work with combinatorics and probabilities, another much simpler way to comprehend the mathematics behind lotteries is to reformulate the problem into a classic urn-and-ball problem:

This urn will certainly not fit the number of balls we need to simulate the probabilities found in a lottery (Photo: MoMath)

What’s an urn-and-ball problem?
Imagine having an urn and some balls of different colours inside. You draw one of these balls randomly from the urn, and its colour will determine the outcome that you get. Suppose if it’s black, you lose but if it’s red you win.

Using a ball to represent each outcome of drawing 6 numbers out of 49, and a colour (e.g. orange) to represent all balls belonging to the same set (e.g. outcomes with 3 matching numbers), we can imagine putting 13,983,816 balls of 7 different colours into an urn, and having to draw one ball at random to determine what outcome we’ll get. Here’s a recap on the number of balls for each colour that we would have to put in:

Grey (0 matching numbers): 6,096,454

Pink (1 matching number): 5,775,588

Brown (2 matching numbers): 1,851,150

Orange (3 matching numbers): 246,820

Purple (4 matching numbers): 13,545

Blue (5 matching numbers): 258

Red (6 matching numbers/jackpot): 1

Do you still think it’s easy to draw that red ball to score a jackpot? It’s actually equivalent to choosing a pixel at random from the plot and hoping that it’s red.

Other Insights

I posted my visualisation on the subreddit /r/dataisbeautiful just yesteday and to my surprise, it generated lots of positive responses (19.3k upvotes!) and interesting discussions about gambling in general.

Some redditors have shared in their comments other ways to put into perspective the irrationality of human decision making with respect to lottery betting.

One mentioned how most lottery players wouldn’t ever consider placing their bets on [1, 2, 3, 4, 5, 6] given how ridiculous it looks, but in actual fact that combination stands just as much chance as any other combination to win the jackpot.

And in order to capitalise on that irrational human behaviour, another redditor suggested that we should perhaps only purchase combinations that are historically rarely purchased by others as that would ensure that the prize pool wouldn’t be split in the event of a jackpot. Sounds plausible?

The Elephant(s) in the Room

Now that I’ve just showed you how the probabilities of lottery function and how they don’t work in your favour, it’s important to point out that that’s just one way to evaluate whether it’s worth playing the lottery.

There’s always a catch… like always. (Photo : Smithsonian Channel)

Another huge consideration that I left out was cost. How much do you pay for each lottery ticket? And what’s the prize money for getting x matching numbers? What’s the expected value of a single ticket? If you want a quick answer as to whether this consideration will be on your side in lottery games, just think again at how many lottery operators there are, and who runs them.

But besides being close to impossible to win, and potentially very costly to play, do lotteries actually have any worth? One interesting aspect to think about is the fun and excitement playing the lottery actually brings.

As one redditor commented, entering the lottery is like, “paying for [the] daydream of being rich”. Perhaps playing the lottery brings value that is simply beyond statistical and monetary measure.

But the question now is, would you ever get exasperated at not getting to realise that “daydream” after having paid for it countless times?