My friend Farida posted a quiz question yesterday, and I responded that the graphic she used looked like a lottery ticket. I suggested that it would be a good question. What are the odds of winning the lottery?
I’m gonna pre-empt that question, by answering it myself, today.
In the UK, you need to match 6 numbers, from a possible 59. That 6:59 format seems quite a common combination globally.
On the night of the draw:
- before you start, there are 59 balls in the pot, and you have six possible numbers on your ticket, so the chance of getting a hit is 6/59.
- Let’s assume you got a hit.
- Second number. One ball has been removed from the bag, but one ball, too, has been crossed off your ticket, so the odds are 5/58.
The odds of getting both the first and second numbers are those two numbers, multiplied, i.e. (6/59) * (5/58). Which is 30/3,422, or 1/115. That’s for getting the first two numbers.
- Now, third ball out. You know the drill.
There’s one less ball in the bag – 57.
There’s also one less number on your ticket – 4.
So, to get just the third ball, you’re 4/57.
And to get all all of the first three balls, you’re
(6/59) * (5/58) * (4/57)
I won’t work that out, because we’re not through yet.
- Fourth ball. Again, one less ball in the bag, one less number on the ticket.
Balls in bag – 56
Numbers on ticket – 3
And, the odds to get all four of the balls so far,
(6/59) * (5/58) * (4/57) * (3/56)
- Fifth ball.
55 balls to choose from
2 numbers left on your ticket.
Odds of getting all five numbers:
(6/59) * (5/58) * (4/57) * (3/56) * (2/55)
- And lastly, the sixth ball.
54 balls to choose from
1 number left on your ticket.
Odds of getting all six numbers:
(6/59) * (5/58) * (4/57) * (3/56) * (2/55) * (1/54)
Phew! So, where are we at? Well, if you got to the very end there, congratulations! You just won the lottery! But let’s finally work out that sum.
(6/59) * (5/58) * (4/57) * (3/56) * (2/55) * (1/54) = about 1/22,500,000.
But it’s not impossible. People do win the lottery, here maybe every month or so.
But more often, nobody wins. On their site at the moment, they say the next draw is a triple rollover – so that is the last two draws, where nobody has won. In fact, winners are sufficiently rare that the lottery company has a rule of no more than five rollovers in a row. After that, they share the prize money out between the people who came closest.
But, would you back a horse at those odds?
My answer, by the way, is “no”. I don’t buy a ticket, which I figure is the most profitable move of all!