So after my post on how dumb it was to play the lottery, a reader emailed me to be more concise with what I meant by “statistically fair shot of winning.” This guy must be a smarty pants because he broke down the math for me. He provided the specific odds of winning the 6/49 lottery. I haz the best readerz on the internetz!

Who is this reader? His name is Eytan. And how does Eytan claim to know all this? He’s an engineer for SkyPro and he, along with his team, created and developed this golfing product that they sell all over North America. I’m going to go out on a limb and guess that he is really, really good at math and knows what he is talking about.

This might be a bit of a snooze fest for those of you not interested in the details of the probability of winning a 1 in 13,983,816 game of chance, so I would suggest you go peruse 15 beautiful pictures instead. Or if you feel like exerting a modicum of brain power, go browse these 10 essential blogging tips for getting more out of your blog.

If you don’t follow my advice, you might risk going all ZzzzZZZZz at work over there. Yes, I know you’re ~~wasting~~ spending time here instead of working! I promise I won’t tell your boss.

However, *IF* you are interested in how the detailed probability of playing the 6/49 works, read on!

I’m just going to copy and paste the email Eytan sent in so as not to ~~fuck up~~ mess up any of the equations he so succinctly laid out. He even put it up on his personal website here. Take it away Eytan!

**ADVANCED PROBABILITY**

Hey Steve,

I love the articles. I’ve started reading them. One article I recommend you touch up is the lottery article. The math doesn’t quite add and might lead to readers losing confidence.

Remember the official odds? 1 in 13,983,816. Statistically speaking, you need to play 13,983,816 times to have a statistically fair shot at winning.

I found this statement a bit vague; what does *a statistically fair shot at winning* mean? I took it to mean you need to play 13,983,816 times to have a 50% chance of winning. If this is what you meant, lets take a closer look at the math and see what we can figure out about the odds of winning Canada’s 6/49.

Lets start by defining some notation. Specifically, we’ll say that p(n) is the odds of winning the lottery at least once after n plays. The odds of winning the lottery for one game is p(1) = 1/13,983,816.

If we played the lottery once, there are only two possible outcomes: win or lose. If you played the lottery twice, there would be four possible outcomes: lose.lose, lose.win, win.lose, and win.win. If we play 10 times, there are 1024 possible outcomes. If wanted to calculate the odds of winning, we’d have to add every single one of those up.

Lets examine the case when we play the lotery twice. As previously mentioned, we have four outcomes and they each having the following odds:

The odds of winning after n plays, p(n) is thus given byAs we play more and more games, the different combinations in which we can win increases, but there remains only a single way to lose. Since the sum of all probabilities is equal to one, we can subtract the odds of losing from one to leave us with the odds of winning.

Now, we are in a position to find out how many times you’d have to play in order to have a 50% chance of winning. We substitute 0.5 for p(n) in the above equation and solve for n.

Wow, that’s a lot of plays. With that in mind, we can clarify the sentence by saying:

Remember the official odds? 1 in 13,983,816. Statistically speaking, you need to play 9,692,843 times to have a 50% of winning at least once.

**CONCLUSIONS**

Well, looks like Eytan is about to put me out of the personal finance blogging business. I should just give him the keys to Kapitalust and pack it up. At this pace, it looks like he could out-math anyone in this blogging niche. It’s a good thing he’s not interested in that (or if he is I choose to ignore him).

All jokes aside, I am amazed that a reader would take the time to break down the probability like this and email me about it. I think it’s incredible. And I had to share it with all of you (who stuck around rather than moving on to look at pretty pictures).

The lesson I have learned: be less vague with my statements with readers like Eytan around.

*Like what you’re reading? It’s time you join the legion of other Kapitalusts! No spam. No ham. No nonsense. Subscribe to receive every new post (and future exclusive offers) via email. It’s super easy to unsubscribe if it ain’t for you!*

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Ugh, probability. I never took a proper probability course, but I did take it as applied to chemistry (chemometrics), so I know a thing or two about it. And I have to admit I’ve never been keen enough to fully write out and correct someone’s math/calculations in long form 🙂

So I think that means you have dedicated readers… good job!

Haha I think Eytan just really, really likes math. Kinda like how we PF bloggers really, really like investing or budgeting or what have you.

I tuned out of most Grade 12 math but probability was one where I really paid attention and loved. Probably because I could see real life applicability of the math.

Now I know why I always failed at probability calculations in high school… They don’t make any sense in my head! I’m glad there are people around like Eytan who can do the math for us. 🙂

Probability was the only section in late high school math that I grasped and understood – I tuned out of everything else…. which led to me almost failing the final exam. Oh, young teenage Steve, so interested in everything beyond academics!

I hear ya Steve! I think probability is the only part of grade 12 math I did decent in, pretty sure I borderline failed everything else. I blame my grade 10 math teacher for making me hate math lol.

I remember in University Economics using grade 11 algebra and getting super excited that I was actually using highschool math for something.

Haha they totally need to re-imagine how to teach math to kids. It’s such a fail of a model to stand there and lecture kids about dry equations and theory – what kid wants to sit through that BS (except maybe Eytan?) 😛

They need to make it applicable and show how it could actually be used in the real world.

Remember that scene in Road Trip at the end when the guy needs to study for a history exam and his friend uses examples from the WWF to make it more applicable and understandable for the guy?

THAT’S what needs to happen with a lot of how school works and teaches kids!

Haha, for me it was definitely the other way around. Geometry and algebra all day every day!

Those weren’t bad, but don’t get me started on trigonometry! Maybe my math teacher just didn’t give good real life examples on how trig applied to real life 😛

I hardly play the lottery myself… I only buy tickets if the jackpot is really, really big! Maybe in some odd chance, I’ll get lucky! I’ve bought local home lotteries for charities before, never won a single thing… the chances were greater too. It’s not my destiny to win 🙁

The only people destined to win the lottery are……..

……….

the people who own and run the lottery companies 😛

A lottery corporation would be my DREAM company to own and run – a ham sandwich could run it and rake in millions and millions of profits!

I was just lamenting the other day that I don’t get many reader emails. Perhaps I should count my blessings.

Haha as long as you have thick enough skin to take constructive criticism with stride, it can be a bit intimidating when someone points out weakness 😛

Well, is he talking about playing within the same lottery multiple tickets? I would almost argue that separate lottery plays are separate events because the winning number changes and playing more may not change the odds. That is because the winning numbers one week are more than likely not going to be the winning numbers the following week. It is not like the normal heads-tails type of odds, you have another variable – that is what numbers actually win.

This is what I thought to, but Eytan explained that probability can seem “counter-intuitive” and that it is a false notion to hold that the odds “reset” after each play.

I’ll email him to ask him to come reply to your comment!

Very observant, Kipp! Separate lottery plays are in fact separate events. Each week when you play the lottery with a single ticket, your odds of winning is always 1 in 13,983,816, regardless of the numbers you choose. However, interesting things start to happen when you consider these events in the aggregate. To borrow your example of flipping a coin, the odds of flipping a head is 50/50, but the odds of flipping a sequence of heads gets smaller and smaller with each successive head.

Oh geez. Glad someone can do this math for us :). Ok, I can now feel even more secure in my decision not to ever buy lotto tickets. All jest aside, I do appreciate a good deep dive on a topic.

Haha I know the feeling because I would have to brush up on the ol’ high school math textbook to break down the probability like that 😛

And I agree that it is good to dive deep into a topic every now and then.

Just to clarify, this doesn’t mean you should NEVER play the lottery, it just means you shouldn’t play when you real return would be less that $13,983,000.

If the prize was 100,000,000 and the odds were 1:14,000,000, I’d be buying a heck of a lot of tickets.

The problem is that large prizes attract large crowds, which I suspect works as a kind of arbitrage to prevent highly profitable lottery situations from occuring.

In any game of chance, if the odds start heavily favouring you should bet hard and go with the odds. Like for a professional blackjack player who waits and waits counting the cards until they know that they are in a position to bet hard with the cards favouring their odds.

However, those kinds of situations are very rare because the people running these games of chance want to win money, not lose 😀

I remember taking probability courses during university. I never liked those courses for some reason lol.

I was just saying above that I LOVED probability in upper high school math because the math and equations were actually something I saw applicable in the real world.

Haha but I don’t actually remember any of it 😛

I think the risk of spending close to $10 million to earn $100 million is pretty good. 50/50 chance right? Just spend $20 million and you might win both times!

If the rewards are sufficiently high, like in your example, I think that could make sense.

However, the odds are to this particular lottery in Canada and the maximum the jackpot can ever reach is $50 million. And it is fairly rare that it ever gets that high.

The lottery ticket for the 6/49 costs $3 to get a single play. So $3 X 9,692,843 = $29 million. You’d have to spend $29 million to have a 50/50 chance of winning this particular lottery. All of a sudden, it doesn’t seem so simple to game the system…

Hmmmm… I wonder if Eytan would have the answer for this question???

Hi Henry,

As my article showed, you’d need to play around 9.6 million times in order to have a 50% chance of winning the 6/49. Checking the Lotto 6/49 website, they list jackpots starting at five million dollars and an individual ticket at three dollars. This means you’d have to spend nearly 30 million dollars in order to have a 50% chance of winning five million dollars.

Not to complicate things, but another scenario would be to purchase multiple tickets. If you only wanted to play once and have a 50% chance of winning the Lotto 6/49, you’d need to purchase approximately seven million tickets. These seven million tickets would be valued at 21 million dollars and the jackpot is five million.

That’s good to know that one cannot guarantee winning the lottery unless every single ticket is bought – which is impossible in a public lottery.

It takes a lot of money to (21 million or 30 million) to even have a shot at 50% in this particular lottery. I think I’d rather take those millions and invest in something that’s a little more of a sure thing 😛

Mmmmmmm. Delicious math.

I love it.

That is all.

These posts make me feel slightly less weird for saying things like “but n equals one!” in conversation. Those conversations are usually limited to my spouse, because most people look at me funny when I say stuff like that.

Haha you need to chat with Alicia, Tawcan, and Eytan more often about complex math equations 😛