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  • We're going to talk about coin flipping.

  • Lets say you got two people. You got Person A Person B, right, they're flipping coins.

  • Let's say one of them is flipping coins and waiting for Heads-Heads to turn up

  • right, so they're going to make a sequence of coin flips, and they're waiting for Head-Heads.

  • Person B, he's doing it and he's waiting for Head-Tails.

  • So one of them is waiting for Head-Heads, and one of them is waiting for Head-Tails.

  • Now, what's the probability that you're gonna get Heads-Heads?

  • Now even Brady would probably know this. What's the probability you get Heads-Heads?

  • Brady: Is it one in four?

  • Yeah yeah exactly. So it's one in four, and Heads-Tails it's the same, right? It's one in four.

  • Even though they have equal probability I claim your average waiting time for Heads-Heads

  • is going to be longer than for Head-Tails.

  • Let's do it, let's do the experiment. Let's not just write out mathematical formula.

  • Let's do it let's go.

  • I'm gonna flip this coin. Now this is, we're gonna do an average. I wanna do a long sequence

  • and we're gonna see what the average is.

  • So if I flip this coin, let's say 50 times, is a nice long number.

  • Then we'll be able to take an average of how long we have to wait for Heads-Heads or Heads-Tails.

  • Right, so we do this if it... actually this is gonna take me ages, isn't it? So instead of flipping this one coin 50 times

  • what i can do is I can flip 50 coins all at once alright

  • which I've just done there, OK?

  • The First coin is a Tails alright. Second coin is, without looking, it's a Heads.

  • Alright, next coin is Tails, I'm gonna a sequence like that.

  • Not even Looking.

  • We've run out of space we better do another.

  • Let's make them random.

  • Right, so I poured them out, I tried not to look at them, so it should be a sequence of random coin flips.

  • I better write out so we can see what they are a bit better.

  • Tail...Head...Tail...Head...Tail...and a Head and a Tail and a Tail...Tail...Head...Tail

  • Alright, we're gonna look at this sequence and we're gonna play the game

  • where tossing a sequence of coins.

  • And we're waiting to see how many, how long we have to wait for Head-Tail.

  • If i start here I'm looking for a Head-Tail, Oh, there it is. Right.

  • Alright that's my first Head-Tail there. That was a waiting time of three

  • and then i got Head-Tail, then I start agina OK. I do a new game

  • I'm gonna look for Head-Tail, and I had to wait one two three four, and there it is.

  • Now i'm gonna do it again, oh look this one is straight away look Head-Tail.

  • Now I'm gonna start again. I'm looking for Head-Tail. Ooh this is a long wait here.

  • And then there it is Head-Tail.

  • And Head-Tail there...there...there it is. Oh and there's one right at the end as well.

  • Right, what was the average waiting time. Let's have a look at the waiting time.

  • So that's about an average wait of 4.5 .

  • Lets look at the Heads-Heads though. You would have to wait, ooh how long? Ooh this is a long waiting time here.

  • It happens all the way over here.

  • There Heads-Heads, which was a long wait. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15.

  • So you've player a game and it ends when you get Head-Heads. Great, done.

  • So now start a new game, a completely different game form this point on.

  • So we'll just start again, oh, but this time notice there the Heads-Heads comes up really quickly. Look at that, it's just straight away.

  • And now we start a new game OK. So we had a success, we'll start a new game and we have to wait 1, 2, 3

  • let's look for our average waiting time for Heads-Heads, OK.

  • Which is 7, right. And yes, oh look, yes.You have a longer average waiting time for Heads-Heads than you do for Heads-Tails

  • even though they have an equal probability, they're both one quarter. Why is this?

  • You can kind of see what that reason is from this, this sequence.

  • Because if you noticed when we were doing it, when you play Heads-Tails

  • you get Heads-Tails, then you start a new game and you wait for Heads-Tails, and you start a new game.

  • Heads-Heads was slightly different, when we did it with Heads-Heads look what happened here.

  • when you have something like this: Heads-Heads-Heads-Heads we have this overlap.

  • And this overlap isn't counted, we were playing it until we got Heads-Heads

  • and then the game stopped. And then we play a new game from that point.

  • So here we've got a Heads-Heads, that didn't get counted as a Heads-Heads.

  • So if you look at how many Heads-Heads we've got it should about an equal number to Heads-Tails

  • but the overlaps don't get counted. Let's just check.

  • How many Heads-Tails did we have? Actually we know we got 11, i know for a fact we got 11.

  • How many Heads-Heads did we have, including the overlaps?

  • One here, we have two, we have three.

  • So if you actually count up the Heads-Heads we had 9 of them, but only 6 were included in our waiting time average.

  • So this sequence, we were looking at what would happen for consecutive values

  • so Head-Heads, Heads-Tails. The same sort of thing would happen with Tails-Heads and Tails-Tails.

  • We're look at consecutive values, in fact the reason I mentioned this is because

  • if you remember there was that prime news where they were looking at primes, and the ending of primes for consecutive primes.

  • This is what they were looking for, they thought if primes were random like coin flips

  • I'm going to find this effect.

  • What turned out to be the case is that they didn't find this effect because the primes weren't being random like coins.

  • If you did this forever, if you had a sequence that went on to infinity the average waiting time then is

  • I'll show what that is. It's called expectation or expected waiting time.

  • The expected waiting time for Head-Tails is equaled to 4

  • Which kind of make sense when it's a quarter probability. That does make sence.

  • And then the expected waiting time for Heads-Heads is longer, and it's 6.

  • So we were close but just a little off. And if you wanted the expected waiting time for Tails-Tails, well that's 6 as well.

  • That's just the same idea.

  • Brady: Audible's ever growing collection of books and other spoken material has now reached

  • in the order of 250,000 titles. So no matter what what you're interested in Audible's bound to have

  • multiple options for you. Now I spend a lot of time driving, as you can see here, often in the UK's

  • delightful traffic and there are few better ways to pass the time than listening to audiobooks.

  • You can have something entertaining, something educational, or maybe a bit of both

  • The choice is yours.

  • I particularly enjoy books about mountaineering. These titles here about Mount Everest are very good.

  • I'll put their details in the description, along with this one about K2 which is one of my favourites as well.

  • And that's just a fraction the the mountaineering books on Audible. I told you they've got something for everyone.

  • If you'd like to give them a try go to audible.com/numberphile and you can sign on

  • for their free 30 day trial. That trial includes your first book and, as I said, there are plenty to choose from.

  • that address again audible.com/numberphile so they'll know you came from our channel, which is handy for us.

  • and then the free trial for 30 days. Our thanks to Audible for supporting this episode of Numberphile.

  • Maths works, maths just works, it's wonderful that way how maths just works out

  • Brady: Every time.

  • Every time just how you want it to work out. Especially with the power of editing.

  • Brady: What happened?

  • What happened, we did out first take and we got the opposite result from what we wanted.

  • Brady: It's done you

  • 48 over 11, which is the opposite of what I said.

  • And we have to film the whole thing again, right?

We're going to talk about coin flipping.

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連續翻轉硬幣 - 數字愛好者 (Consecutive Coin Flips - Numberphile)

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    林宜悉 發佈於 2021 年 01 月 14 日
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