13,983,816 and the Lottery - Numberphile

JAMES CLEWETT: Today's video is going to be all about the National Lottery. And if you're a number freak. like me, then the National Lottery is great because it's, that two times a week when all the people at home get, to share in my enthusiasm for numbers. So what I want to do today is. talk a little bit about some of the probabilities involved, in the National lottery. So before we start, I think we should choose some numbers for our lottery. So our first number is going. to be the number 14. That's my birthday. Then I think we'll have the number 19. That was a nice, good year, for me, 19 years old. We'll have a few in the 30s-- 31, and 33, and 34. OK, and then one more, 45. So I've chosen the number 14 for my birthday, the number 19, and then, a few in the 30s,, number 31, 33, and 34, and number 45. So now, what are the odds of us picking those same six balls out of our homemade, lottery kit? First of all, to start with,. there are six balls in here which are winners for us. And there are 49 balls, just like the national lottery game. OK, so I'm going to look away. And I'm going to select. And it's number 19,. a 6 out of 49. OK, so we put that. one to one side. But now they're only 48 balls left in the basket. And we've got five winners, left in the box. OK, let me give that, a good stir around. And it's the number 45. Put that one to one side, and, again, now, we've got one less ball in the box. So we're down to 47 balls and, just four winners now. Give them a good stir. It's the number 31. And that one, again, goes to one side with our other winning balls. Just 46 balls left in here. now, and only 3 winners. There it is. The number 14. OK, again, put that, one on one side. Give ourselves a little stir. Now there's only 45 balls left,

and just two winners. There it is. The number 33. OK, so we've got one winning, ball left in the bucket. So give it a stir. There it is. The number 34. OK, so my chances of picking out the number 34, were just 1 from 44 balls. And there we have it. What were the odds of me, winning the lottery and retiring to become a part-time, physicist? Well, let's have a look at it. The chance of getting the first, number was 6 in 49. Then it's a completely. independent event, so we can just multiply that by the chance, of getting the second number, which is 5 in 48. So I can just just multiply that. Then we multiply that, by the chance of winning the third number-- the top line we can. do in our head-- is simply 720. The bottom line-- I'm going to cheat-- I'm going to use a calculator. 49 times by 48 times by 47 times by 46 times by 45 times by 44. And that's the number 10, billion and change. So we're going to write down, our 10 billion here. So that gives us our final, answer in math speak, if you like, of 7.151 times 10 to the minus 8. That number doesn't really mean anything to most of us. And so I'm going to write, this as 1 over 13983816. So the odds of winning the lottery are one part in 13,983,816. Or in real money, about one, part in 14 million. I play the lottery at Christmas. Actually, I find it's been, really good fun. So I tend to spend about a fiver a year on lottery numbers. And for that one week that. I've bought a ticket, I daydream about all the really, cool things that I'm going to do if I win the lottery. So how much is that worth? I don't know. I really enjoy it. I really enjoy just the process of owning a lottery ticket. But I know-- I'm reasonably numerically. grounded-- I know that my chances,

of winning are very, very close to zero. I think my chances of being, struck by lightning are something like seven times higher than my chances of winning the lottery. So I'm kind of hoping. that doesn't happen. So we've won the lottery. We've matched six balls. That's absolutely fantastic. But it doesn't really explain why every time I've bought a lottery ticket, I've not matched, any balls at all. So what's the probability of. buying a lottery ticket and matching no balls? So let's go back to the numbers that we selected. So we had 14-- OK, so this were our magic numbers. So what are the odds of, selecting a first number which doesn't match? Well, there are 43 balls. in there, out of 49 which don't match. So I'm going to select one. It's number 38. OK, so that doesn't match. I don't want that one. Give it a stir. Now remember, here we go. That's the number 35. And, again, it doesn't match. Give them a stir. It's the number 25. Again, no use to me. 41 out of 47. OK, so the next one. Number 23. Getting a little annoyed, with this. That's no good to me either. So the chance of pulling out a nonwinner there were 40 out of 46. OK. Do another one. I've got the number 41. No use. OK. So the chances, again-- 39 out of 45. And our last ball-- BRADY HARAN: Hang on, what do. you think it's going to be? JAMES CLEWETT: Well,. I don't think it's going to be a winner. That's for sure. Uh, give it a stir. I hope it's not a winner. Number 43. OK, so we've got no balls at all. Rubbish. Right. The chances of a nonmatch-- 38 out of 44. If we calculate this number. through-- again I'm going to turn to my calculator. OK? So our probability of picking, no matches whatsoever is 4 billion divided by 10. billion, which is-- sorry-- 43%, 43.59649%. Somebody is spending a pound, a week playing the lottery. And for that pound a week, they're getting hope and excitement and a Saturday. night buzz. I think that's great. value to be honest. So that's fine. But if I hear of somebody who's. spending 10, 20 quid a week buying lottery. tickets, stop. Please stop.,