Placeholder Image

字幕表 動画を再生する

  • I'm gonna show you a little bit of a scam, or a trick, that you can try out on people.

  • So, what I'm going to ask you to do, is to pick a sequence of three coin tosses.

  • So, you know, heads-tails-heads, or tails-tails-tails.

  • Brady: "Tails-tails-heads."

  • Okay. So, Brady's picking tails-tails-heads.

  • And I

  • will pick heads-tails-tails.

  • And the game is: we're gonna toss this coin and see who gets that first.

  • So, let's see who wins. We'll do a best of 5.

  • Okay? Something like that.

  • Okay, so,

  • let's try game Number 1. All right, here we go.

  • So, number 1...

  • All right, let's write down this sequence.

  • Heads.

  • Okay, that's pretty good actually; that's good for me.

  • Yeah, it's good for me, not good for you.

  • All right, heads again.

  • Heads again.

  • Ooh, look at this.

  • We might have to do a different video about how many heads we can get in a row.

  • Oh look, tails. Alright

  • Oooh, tails.

  • That is a win for me. Excellent.

  • A James win, brilliant.

  • Alright, well, best of 5?

  • Alright, Brady's got a tail. Come on.

  • Brady: "Ah, now, this is good for me."

  • Yes! That's a tail.

  • Brady: "Yes!"

  • - And...

  • It's a tail!

  • Oh no you're okay though

  • You're still, you're still in the lead

  • Yeah, happy with that aren't you?

  • Alright, oooh

  • Brady: "That's good for me."

  • Tails, tails

  • Brady: "In fact, I'm definitely gonna win this next one"

  • Soon as the sequence breaks, yeah,

  • you're right.

  • Well observed, Brady.

  • Brady does win that, he had to win that after all those tails coming up. Okay,

  • Brady: "So, It is bad to choose any two consecutives as you start."

  • It is bad to choose.

  • Brady!

  • Brady: "I realize what I did wrong."

  • Ahhh cmon cmon cmon you can still win this

  • Have you given up?

  • Brady: "I've realized my mistake."

  • You've realized what a mistake

  • You're less likely to win than I am.

  • Brady: "So, what would you have done if I had not been stupid and not given consecutives as my..."

  • Whatever sequence that you pick

  • I can pick a sequence that is more likely to appear before yours.

  • Brady: "What if I pick the most likely of them all?"

  • Exactly, what if you pick the most likely of them all?

  • It goes round in a circle like rock, paper, scissors

  • Let's say that Brady picks whatever sequence he does pick

  • I'm going to show you what sequence I should pick so that I can beat him.

  • Right, or at least with a better chance of beating him.

  • I'm going to show you the cycle of winning. Let's do that.

  • Right. So let's say Brady picks heads-heads-heads, right? Three heads in a row.

  • Then what I should pick is tail-heads-heads.

  • And tail-heads-heads will beat heads-heads-heads.

  • If Brady had picked tails-heads-heads,

  • then I should choose tails-tails-heads.

  • Because that is going to beat Brady.

  • If Brady had picked tails-tails-heads,

  • then I should pick heads-tails-tails, that will beat Brady.

  • Incidentally, if he had picked, instead, tails-heads-tails,

  • you pick this one here, tails-tails-heads. So it's not quite a circle, this.

  • It has some spikes coming off the side of it.

  • If Brady had picked tails-tails-tails, you want to choose heads-tails-tails.

  • If Brady had picked heads-tails-tails,

  • then the best choice is heads-heads-tails.

  • If you had picked heads-tails-heads, I would again pick heads-heads-tails.

  • If you had chosen heads-heads-tails, Brady, I would have picked tails-heads-heads.

  • So we have this little circle, but you can see the spikes coming off it as well.

  • So to read this, Brady's choice would be here, at the pointy end of the arrow.

  • My choice, the better choice, would be here, so the arrow going towards Brady's choice there.

  • There is a way to help you to remember this cycle.

  • The way you do it is, when your opponent picks a sequence, let's say, tails-heads-heads,

  • what you should do is, in your mind, copy the middle one, so make a copy of that coin,

  • so I've got another heads there, I'm gonna put it at the front, and flip it over.

  • So it would become a tails. And then my choice would be tails-tails-heads.

  • That's the winning choice.

  • That little way of remembering it is better than just remembering that cycle.

  • And that will work. That's your best choice to go for.

  • I should show you the probabilities for each choice.

  • If Brady picked heads-heads-heads, I would pick tails-heads-heads.

  • And the chance of beating you is actually pretty good. It's seven eighths. I can tell you that's round about 87.5 percent.

  • So it's really likely. I'm really likely to win the game.

  • That's why I did best of five, though, just in case probability let me down.

  • This probability here, if I wanted to work tails-heads-heads beating heads-heads-tails,

  • that happens with a chance of three quarters, so it's about 75 percent,

  • really big probability.

  • If you wanted to do this one here, heads-heads-tails beating heads-tails-heads,

  • that happens with a chance of two thirds, so that's, what's that, about 67 percent.

  • And then, actually the others are similar.

  • This probability here is two thirds, this one here is seven eighths, this one here is three quarters.

  • There's a symmetry in this.

  • So you are far more likely to win.

  • Excellent. Try it out on your enemies, right. So you can beat them.

  • This is insane, because you think, well, surely there's a best choice.

  • There's a best choice, and then, all the others are worse than that.

  • And you can't beat the best choice.

  • It works in such a strange way that it makes this cycle of probabilities instead.

  • Just to show you where these come from. These are some of the easy ones, okay?

  • I'm gonna show you where some of these come from. Well, look, this is an easy one, look at this.

  • I said this was seven eighths. Yeah, that's actually an easy one to spot,

  • because, well, you could get heads-heads-heads straightaway, which happens one eighth of the time.

  • But if you don't get heads-heads-heads straightaway, if you've got a sequence,

  • and somewhere in the sequence it's heads-heads-heads, let's say this is the first appearance,

  • then it has to be preceded by a tail.

  • If it's not, if it was preceded by a head, then it wouldn't be the first heads-heads-heads in the sequence.

  • It has to be preceded by a tail.

  • Which means it has to have tails-heads-heads coming up before it

  • unless

  • you get the three heads in a row straightaway.

  • So, yeah, you're going to lose if you pick heads-heads-heads, most of the time.

  • Yeah, so we go back to our game, what Brady chose. He chose tails-tails-heads, it wasn't the best choice.

  • Brady, if you picked tails-tails-heads, I used my little algorithm, I know what I'm supposed to choose,

  • heads-tails-tails, and I can beat you, appear before yours does, with a chance of 75 percent.

  • Which is actually the second best thing there.

  • Really bad choice, Brady. Sorry about that.

  • Well I might not be making the best coin choices, but I have been learning a lot of new mathematical tricks

  • at TheGreatCoursesPlus. This is a great online resource for anyone who wants to learn anything.

  • Become smarter about things from cookery to quantum mechanics.

  • I think numberphile fans, in particular, might be interested by some of the mathematical offerings.

  • They're really extensive, and right up the alley of the sort of people who watch these videos.

  • There's lots about games, puzzles, probability. I've really been enjoying, just today, some of the videos about probability.

  • I wish I'd watched them before recording that video with James.

  • If you'd like to find out more, go to TheGreatCoursesPlus dot com slash numberphile.

  • Have a look what's there, and if you like it you can actually sign up for a one-month free trial.

  • That's one month's access to 7000 plus videos, all taught by leading experts in these fields.

  • That address again, TheGreatCoursesPlus.com/numberphile.

  • Give it a look. If you like these videos here on numberphile, I think you might really like what you find there.

  • And cheers to the GreatCoursesPlus for supporting this video.

  • This game is called Penney's Game, or penny ante, which I think is a kind of a pun.

  • Actually, the strange thing is, Penney doesn't refer to coins and pennies, and flipping coins,

  • It's actually the name of guy who came up with the game. He was called Penney.

  • It's kind of one of those situations when you have a baker called Mr. Baker,

  • Or a mathematician called Dr. Sexy or something like that.

I'm gonna show you a little bit of a scam, or a trick, that you can try out on people.

字幕と単語

ワンタップで英和辞典検索 単語をクリックすると、意味が表示されます

A2 初級

ペニーのゲーム - 番号マニア (Penney's Game - Numberphile)

  • 4 0
    林宜悉 に公開 2021 年 01 月 14 日
動画の中の単語