Understand Your Chances Of Winning The Lottery In 5 Minutes Flat
Most lottery players don't really understand their chances of winning the lottery.
Because normally, understanding it all takes more maths than most of us ever did at school. And more thinking than we do in an average day at work!
So no college degrees needed for our explanation!
Table Of Contents:
Starting With Toast
We won't bore you to death with statistics and formulas - you hate that right? So let's keep it fun, OK?
You're making breakfast. And as a result of your half open eyes and carpet slipper shuffle - you trip - catapulting your toast across the room..!
So, what is the chance it will land 'butter side down'?
OK, so assuming 'sods law' is not at play in this particular universe, there are 2 sides the toast can land on. I think we can safely agree then that it's a '1 in 2' chance of butter side staining your carpet...
This time you were 'lucky', it was dry side down.
That's lesson 1 done. Relieved..?
Picking Your Toast Up...
You bend, pick your toast up, blow most of the hair and grime off it - and return it to your plate (why not, 5 second rule..?).
Just as the dog barrels into your legs on route to a serious barking at the mailman. Your bleary eyed dismay watches your toast take off for a second time...
So what are the chances it will stain your carpet this time?
If you said '1 in 2', well done, you've already got lesson 2 down.
(And this is way more important than you might think)
Not sure why?
There are still 2 sides to your toast. And still only one of them has some butter on it. That means there is still a 1 in 2 chance of a grease mark on your carpet!
The fact your toast fell 'dry side down' the first time will NOT make it any more or less likely that it will be butter side this time.
Clearly it's your lucky day, it was dry side again. Just be thankful you didn't choose cornflakes today.
So, previous results do not affect the future. That's lesson 2. Easy stuff this probability theory!
What's This Got To Do With My Chances Of Winning The Lottery?
OK, let's start talking balls ("you already were...").
If there was a really dumb lottery that had only 2 balls. And they drew out just 1 of them to pick the winner. What would your chances of winning be?
Yes, it's the toast example in glorious lottery 3D. Which means, your chances of winning are 1 in 2.
And of course, it means if ball 1 was picked last week, the chances of it being picked this week are just the same. Ball 2 didn't suddenly become more likely because it didn't come out last week.
More Balls Please!
Right. So what happens if we add another ball?
Pretty easy right? Our odds now change to '1 in 3'. Because we are still picking out 1 ball, but there are now 3 balls to pick from.
OK. So what if we draw 2 balls out of 3...
Ouch. That got harder. It sounds like it might be '2 in 3' - but it isn't.
Just to prove it to you. If you think about it a bit more (don't worry, it's nearly time for a coffee break!), here are all the possible combinations that can be drawn:
| Ball 1 | Ball 2 |
|---|---|
| 01 | 02 |
| 01 | 03 |
| 02 | 03 |
| 02 | 01 |
| 03 | 01 |
| 03 | 02 |
But wait a second. Did you spot it? When we're talking about your normal lottery draw, it doesn't matter what ORDER the balls are drawn in, so long as they match the ones on your ticket - right?!
So, if you look a little closer at those numbers in the table, you can see that 01, then 02 is exactly the same result as 02 then 01! If your numbers were 01 and 02 you'd hit the jackpot for either of those.
So there are actually 2 ways of drawing every result, 01 first then 03 - or 03 first then 01 etc.
Confused yet? Waddya mean 'no'! ;-)
You have now found the secret formula for working out your chances of winning the lottery!
No, really! You have.
You worked out there were 6 possible ways the balls could be drawn. And you worked out that there were 2 ways each set of results could be drawn - because it didn't matter what order the balls were in for you to win.
The only thing you didn't do was to divide the 6 possible results by the 2 ways of drawing them. Which gives the correct answer - a '1 in 3' chance of winning.
You may now call yourself a student of lottery statistics and probability. Nice one.
Even More Balls!
Let's get dangerous. We're going to jump up to a 4 number lottery with 3 balls drawn. Ooooooh. (Patience eager student, 49 balls shall come to those who wait!).
OK. So if you write down all the possible combinations that can be drawn, how many are there?
Oh, alright, I'll do them for you...
| Ball 1 | Ball 2 | Ball 3 |
|---|---|---|
| 01 | 02 | 03 |
| 01 | 02 | 04 |
| 01 | 03 | 02 |
| 01 | 03 | 04 |
| 01 | 04 | 02 |
| 01 | 04 | 03 |
| 02 | 01 | 03 |
| 02 | 01 | 04 |
| 02 | 03 | 01 |
| 02 | 03 | 04 |
| 02 | 04 | 01 |
| 02 | 04 | 03 |
| 03 | 01 | 02 |
| 03 | 01 | 04 |
| 03 | 02 | 01 |
| 03 | 02 | 04 |
| 03 | 04 | 01 |
| 03 | 04 | 02 |
| 04 | 01 | 02 |
| 04 | 01 | 03 |
| 04 | 02 | 01 |
| 04 | 02 | 03 |
| 04 | 03 | 01 |
| 04 | 03 | 02 |
OK. So there's 24 combinations. Now look closely, like before. How many times is each set of numbers repeated? How many times, for example, can you see balls 01, 02 and 03 but in any order?
I make it six times.
So with your magic formula, that means we divide 24 possible combinations by 6 ways to draw each set of numbers.
Which gives us 4. Or a '1 in 4' chance of this jackpot.
Look What Happens When We Add One More Ball
So let's still draw out 3 balls. But now from a total of 5 balls.
Even I'm not going to write all those combinations down. But I will let you in on a nifty way to discover how many there are, for any number of balls.
All you have to do is this: 5 x 4 x 3 = 60 ways.
Eh, how did that work!?
Easy. You start with the highest numbered ball, in this case 5. And keep multiplying by the next smaller numbered ball. For as many balls as you are drawing out.
So, if you were drawing 2 balls out of 3, like our example we did earlier. Well that's just 3 x 2 = 6. Which is exactly what we worked out, the long way!
Or, for 3 balls out of 4, it's 4 x 3 x 2 = 24, which is also the same as we worked out above.
Nifty isn't it!
But How Many Combinations Are There In A Real Big Lottery?
Most of us play a lottery with 49 different balls, and 6 balls get drawn.
Everything works the same way. Just be ready for some big numbers!
We just do the same as before: 49 x 48 x 47 x 46 x 45 x 44 = 10,068,347,520.
That's a big number and an awful lot of combinations. But don't forget that it doesn't matter what order the balls are drawn in!
Here's our second big time saver.
It would take forever just to write all these combinations down. Then trying to figure out how many times combinations are repeated would be a very long and dull day.
When you can simply do this: 6 x 5 x 4 x 3 x 2 x 1 = 720.
You can maybe spot where that little shortcut came from.
It's 6 balls to be drawn, multiplied by 5 left to draw, then 4 left etc.
But let's double check this against our answer above. When we drew 3 balls we worked out the combinations were repeated 6 times didn't we? Which is 3 x 2 x 1.
Easy when you know how.
That lottery statistics diploma is nearly in the bag!
So What, Then, Are My Chances Of Winning The Lottery?
Here's where the magic finally happens. Because you've pretty much already worked it out.
We just take the big number, and divide it by the smaller one! Or in other words divide the number of different possible combinations, by the number of ways they can be repeated.
So it's 10,068,347,520 divided by 720.
Which comes out to '1 in 13,983,816'.
And that's why you don't win the jackpot every week!
Well done for staying awake this far. If you've had less than 3 cups of coffee, I'm impressed!
How To Increase Your Chances
So now you know how tough the odds are, I suppose you now want to know how to make it easier to win?
One way is to play less often but play the same number of tickets/combinations overall.
That's right, 10 combinations played in a single draw have a better chance than 5 combinations played in two draws. Same cost to play, but slightly better odds!
That might make your head hurt just a little. The math certainly will so that's one for another day :-).
Just bear in mind it is only slightly better. And playing once a year is definitely not nearly as much fun as playing once a week. So don't go crazy on this one!
The other (obvious) way to increase your chances is... buy more tickets.
It's easy to see that if you buy two tickets, you double your chances. So that 1 in nearly 14 million, becomes 2 in 14 million instead (because you bought 2 of the 14 million possible combinations). Which is also the same as 1 in 7 million.
Wahay, now you're gonna win big. ;-)
Don't forget though, that there are also prizes for matching 5 numbers out of 6, or 4 numbers. Or even just 3.
So your chances of winning the lottery aren't quite as bad as they look. Still pretty grim for the jackpot though!
But the fact is that even just one ticket has an infinitely better chance than no ticket at all :-)
Congratulations on becoming an Expert in Lottery Statistics and Probability.
