### Coin Game \ Magic Trick \ Maths: The Penney Ante Part 1 (Re: Derren Brown: How to Win the Lottery)

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**Theodore Winn** - 4 weeks ago
- 47 comments

Hello, everybody! So–about last night. For people not in the UK, last Wednesday, a brilliant UK magician called Derren Brown predicted the national lottery numbers on live TV. And last night he explained how he did it. Now, I won’t go into that, because Derren gave a few explanations, and they were all nonsense. However, I did laugh when the following coin game came up as proof of psychic ability. Now, Derren did say that this was a maths effect, based on what he called “deep maths.” I’m going to explain that game. I’m going to explain that game, I’m going to tell you the optimal strategy for you to win, and I’m going to explain some of that “deep maths” behind it. And hopefully you’ll be able to try this out on your friends at home. Now, the game is called “Penney Ante.” It was devised in 1969 by a mathematician called Walter Penney. It works like this– player one goes first. He has to predict three coin tosses. So these are the possibilities. It could be heads-heads-heads, heads-heads-tails, heads-tails-heads, heads-tails-tails, tails-heads-heads, tails-heads-tails, tails-tails-heads, or tails-tails-tails. So there are eight possibilities. Player one picks one of these as his prediction. Player two then makes his prediction, and the prediction that appears first in a run of coin tosses wins. Now, since these are equally likely, it sounds like a fair game. But, if you go second, there’s a strategy to help you increase your chances of winning. For example, if player one picks heads-heads-heads, player two should pick tails-heads-heads. Because if this is a run of coin tosses– let’s say this is the first time heads-heads-heads appears in this run. Then the preceding coin must be tails. If it was heads, then this wouldn’t be the first time heads-heads-heads appears in that run. So the preceding coin must be tails, which means player two’s choice, tails-heads-heads, must appear first. And player two wins! Now, the only exception to this is if the first three coin tosses are heads-heads-heads. Now, the chances of that happening are one in eight, so the probability that player two is going to win is seven in eight, which are pretty good odds! Now, given player one’s choice, this is a table that tells you what player two should do. For example, if player one picks heads-tails-tails, then player two should pick heads-heads-tails. And if he does, he’s going to win with a probability of two-thirds. That’s about 67%. Now, there’s an easy way to remember this table. What you do is you take player one’s choice, you make a copy of his middle coin, and you flip it over, and you put it at the front. So if player one picks tails-heads-tails, you make a copy of the middle coin, heads, flip it over so it’s tails, put it at the front. And so player two’s choice is tails-tails-heads. And if he does that, the probability that he will win is three-quarters– 75%. Pretty good! Now, this is where most explanations stop. After the break, I’m going to go a bit further. I’m going to show you how we work out the best strategy, and those chances of winning.

hello everybody

You try operating the camera, reading the autocue and presenting a 12 minutes maths video without mistakes.

Man ur smart 😀

There's always something. I can not described how annoyed I am about that.

wow smart

What Darren Brown did was amazing! He is one of the best! Also, this is a cool "game" I might just have to use it…

It was amazing. I have full respect for him as a magician. He often uses maths tricks in his act as well, dressed up as mentalism.

i saw that he doesnt really explain it the coin game is awsome! i can do it he told how to do it it rocks! u live in the uk?

Then I did well 🙂

To be fair to Derren he didn't have the luxury of being able to spend 12 minutes on it as I have. In TV terms that would be deep maths 🙂 Glad you didn't think so.

As for the other thing, I already have a job thanks 😉

WHAT???

Everyone been talking about the Derren Brown thing. Ask magician Kevin Cunliffe (on my FB friends) what he thinks of it. He told me in the pub on Saturday but I forgot what he said (he was busy being pestered by people asking him to bang a nail in his nose as usual). I think he's met Derren Brown.

loving your work !

Friends at home lol…

lol, I thought that at the time "deep maths"?!? I'm in GCSE maths set and I worked this out in 30 seconds. Come on…

Swap the two columns that say Player 1 and Player 2. If Player 1 chooses HHH the best strategy is THH, which will win 7/8. But on the other hand, if Player 1 picks THH then the worst strategy would be to pick HHH, which would win only 1/8. However, unlike the winning strategy, the losing is not unique. There can be more than one choice that would be equally bad.

That's it, that's the full table. And for each choice by Player 1 you just need to go down the column and pick the smallest probability to find the worst choice for Player 2. Good question!

in cambrige what math cours do you teach?

Do you teach math with physics?

nice video I learned alot.

haha im a junior in highschool, you should help me with my pre calc lol

looks a bit like john lydon!

i wonder if this would work on poker machines with the colours of red and black

@singingbanana could this strategy be used with the red/black option?

@hyperm0d Turn on your annotations. The last four are in the wrong order. My mistake.

Makes absolutly no sence. The chances are 1/8 no matter what.

@srtyhtm Wrong wrong wrong. Its simple logic. Easiest one to realize this on is the very first one.

HHHvsTHH

Unless the very first 3 flips are HHH, then player 2 will win no matter what. It impossible for HHH to win if a single "T" ever appears. Because anything heads in a row only add to THH's win.

This was featured on scam school awhile back, a very nice mathematical trick.

The prediction that appears first? Doesn't that mean that player one wins every time? Please explain a bit clearer, I didn't understand anything of the concept of the game.

@ANGRYHILLBILLY2 I do that sometimes.

this guy's so cool that youtube advertisments do not appear on his videos 😀

The 'way to remember' doesn't work… Help!

He reminds me of Johnny Rotten somehow…

Seriously… this guy… probably the most under estimated youtuber ever… I enjoy watching his videos more than anyone elses

man you are brilliant!!! i enjoy watching your videos on mathematics!!! especially this one, it helped me understand a concept in probability..

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where did you learn this? from others videos? or at school? tell me you secret 🙂

Come to Sweden plix? kkthxbai

I wanted to do this on a friend. he always chose first but he always won. We did it like 4 times. Luck is a b*tch.

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You should like yourself, but it's bad form to like yourself online.

Hello from Arizona. Today's was my first Math Contest with 20 questions in an hour!. I am a very fan of you and all you videos including numberphile. So one of the question was THIS question, the probability to win if player one choose HHH. Because of this video I was able to answer without doing the probability process. Thank you!!

Derren Brown did NOT predict the lottery numbers. A prediction is made before the event, Derren showed his numbers the draw and he used a split screen to accomplish the "illusion".

@schwarzeneggermagic anyone?

@srtyhtm It's all in the presentation. The idea is that when your sequence shows up, there is a 50% chance that I

~~-HAVE ALREADY WON-~~, and then I "start a new round" in order to avoid you collecting any points.Derren started over on a new line, so 50% of the time his opponent's sequence showed up, Derren didn't count it:

You pick THH, I pick TTH. My combination is surrounded by parentheses.

TT(TTH)(T*TH)H*. Note that you also won at the end (starred), but the way Derren would have written it was:

TTTTH

TTH

H…

This makes the opponent not realize that the last THH combination would (should) count.

I came here from the numberphile video, and funnily enough his explanation on both the original and "extra" videos doesn't go beyond what he said here. Yet, this video ends with "most explanations end here, and I'll go deeper in part 2." Do that part in numberphile dammit!

I was looking at the table but if player 1 choose TTT en player 2 chooses HTT. Wouldn't the probability of player 2 victory be 7/8 just like by a game where player 1 chooses HHH?

I explained this to a friend and still he lost thrice in a row after all the advantages. He is asking for explanations. Except your HHH example where a pattern, HHT…..HHHTT must have a T before H, are there other examples to explain this?

HHH – THH – 7/8

HHT – THH – 7/8 https://en.wikipedia.org/wiki/Gambler%27s_fallacy – (NOT 3/4 – This is the Gambler's Fallacy).

(T)HTH (1/16) – HHT (1/8) – 2/3 – Player 1 needs four correct tosses, player 1 only needs three.

(T)HTT – HHT – 2/3 – Same thing…

Then repeat inverted.

where is part 2?