My friend Farida posted a quiz question yesterday, and my considered response was that the graphic in her question looked like a lottery ticket, and I suggested that she should one day ask, what are the chances?
I’m gonna pre-empt that question, by answering it myself, today. In the UK, we have a lottery where you need to choose 6 numbers, from a possible 59. That 6/59 format seems quite a common combination globally.
So, let’s assume you chose your six numbers (their name is a line).
On the night of the draw:
- before you start, there are 59 balls in the pot, so you have one chance in 59 of a particular number coming out of the hat. But you have six numbers, don’t forget, so the chance of the drawn number matching any of the numbers in your line 59:6
- Now, let’s assume you got a hit.
- It preparation for the second number, there is one less ball to pick from the hat, there are only 58 balls. But equally, you have one less number in your line, because you already crossed one off. So the chance of a particular number coming out of the hat is 1 in 58. But remember, you have five available numbers, so the odds of matching one of them is 58:5.
Now, at that point, let’s take a timeout. What are the chances of both the first number coming out, and the second number coming out?
Well, we know that the chance of the first of these is 59:6, and the chance of the second is 58:5. In fact, to work out the chance of both these things happening, you need to multiply those two numbers, so the chance of both balls being hits is
(59 x 58):(6 x 5), which is 3,422:30. That’s about 115:1, that’s how much chance you have of matching two numbers. That’s for two hits, so let’s keep going.
- Now, there are only 57 balls in the pot, so the chances that the next ball drawn will be a particular number is 57:1. But you still have four numbers to choose from.
- So, the odds of matching all three balls so far? You got it, they are (59 x 58 x 57) : (6 x 5 x 4). I won’t bother working that out, because we’re not finished yet. And it’ll make my head explode. Let’s work it all out at the end. Let’s say you match this one too.
- Let’s move on to the fourth ball. Now, there are only 56 balls in the pot, so the chances of the next draw matching one of your numbers is 56:1. But you have 3 numbers, so that makes the odds 56:3.
- And, you’re gettin used to this, the chances of having matched all four balls to this point are (59 x 58 x 57 x 57 x 56) : (6 x 5 x 4 x 3). But let’s assume you keep matching the numbers.
- Fifth ball. One chance in 55 for any given number. The number of balls is going down by 1 each time. But now, the draw has to be one of two numbers. So the chances of this ball matching one of yours is 55:2. And the chances are getting all five hits is (59 x 58 x 57 x 56 x 55) : (6 x 5 x 4 x 3 x 2). Nevertheless, assume a hit.
- Sixth ball. One chance in 54 on a particular ball coming out. But you only have one ball, because the other five numbers were all hits, and are therefore out of the game You have Chances of getting all six numbers are (59 x 58 x 57 x 56 x 55 x 54): (6 x 5 x 4 x 3 x 2 x 1).
So, where are we at? We started off with 59 numbers in the pot, but you had 6 numbers in your line. Each time we draw a number, the pot gets smaller by one. /but at the same time, you cross another number off in your line, so it gets shorter by one. And it we end up crossing off all six numbers in our line, we just won the lottery! We’re millionaires! There are, of course, prizes for matching fewer numbers, But the chances of actually winning the thing are more than 1:22, 5oo, 000. Would you back a horse at those odds?
That’s not as impossible as it seems.People here do win the lottery, maybe every month or so. But more often, nobody wins. I presume it are pretty typical at the moment, and if I go to the lottery web site, they are adverticing the next draw as a triple rollover – so that is the last two draws, where nobody has won.
Now, people in the UK have gradually seemed to lose interest in the lottery, and I’d guess that ticket sales have fallen over time. The lottery company is aware of this, so they also have a rule of no more than five rollovers. After the fifth, they share the prize money out between the people with the most matches.
Did you get all that? Beware, I might be asking questions! But judge for yourself – could it happen to you? (Not to me – I don’t buy a ticket, which I figure is the most profitable move of all!)