Coin Game \ Magic Trick \ Maths: The Penney Ante Part 2 (Re: Derren Brown: How to Win the Lottery)

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Penney Ante is a coin game devised in 1969 by a mathematician called Walter Penney. In it, two players predict three coin tosses. The prediction that appears first in a run of coin tosses wins. This should be a fair game. However, if Player 2 makes his prediction after Player 1, he can use mathematics and probability to increase his chances of winning.

On Wednesday 9th of September 2009 (09-09-09) UK magician Derren Brown predicted the national lottery numbers on live TV. The following Friday he explained how he did it. This included a demonstration of the Penney Ante.

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You're right, and thank you for pointing it out. For the balls my method was far more complicated than it needed to be, I was just demonstrating the rule we were about to use. For the coins, this is the best method of which I know.

singingbanana
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Permutations are ways to order objects, so HHT HTH THH are different permutations of HHT. More on that in the next video.

singingbanana
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@mccahill124 I hope you'll keep watching then. Just taking an interest shows you in a good light!

singingbanana
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In the coin flip race between HTH and HHT (fair coin), their individual 'wait times' are 10 and 8 (10 = p^-2*q^-1 + p^-1 and 8 = p^-2*q^-1) the average number of coin tosses is 6 (mean length of a game). (from a Markov chain and a variation of John Conway's algorithm). Now, GIVEN a HTH win, the mean number of tosses = 5 1/3. For HHT it is 6 1/3 (using a recursion method - from a Markov chain and verified with simulation). HTH wins faster - when it does win, on average, than HHT. The Penney-ante game has LOTS of non-intuitive facts in almost every 2 player race.

kraps
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I believe so (I may have remembered that wrong, but I think that's what it was). The first hit on a google search for Penney Ante is a teachnical paper for the optimum strategy in general. I'll send you the link too.

singingbanana
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He is good. I will be watching. The week after is gambling, which sounds quite similar to the lottery show - in that there will be some maths in it. I think Derren quite likes the maths, he uses it a lot, to brilliant effect.

singingbanana
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The first video is the extreme case. It isn't obvious in the other cases until you do the maths. However that is where most expanations on the web stop.

But if you look at the large table in part 2, a lot of player 2's choices just give him a 50% or worse chance. You have to do all this maths before you find the strategy. The idea of flipping the middle coin isn't significant, it's just a way to remember the best choice. And for four or more coins the way to remember best choice is different

singingbanana
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If the second player follows this stratergy:
If 1st player chooses HHH P(2nd wins)=7/8
If 1st Player chooses TTT
P(2nd wins) =2/3
But are these cases not symmetric??? Shouldn't the TTT case also have P(2nd wins) =7/8???

purushotamgarg
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You are right, I have already put an annotation in part 1, but the same is tue here. But very well spotted.

singingbanana
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I try to be as clear as possible, but if I could I would make some changes. Dash it. The end result is pretty cool though.

singingbanana
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Isit me or is there a buzzing sound in e video

LosDynasty
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I think the mnemonic question is a good one. When reading about it, I got the impression that the flipping the middle coin bit wasn't something significant, or insightful.

For n>3 I think you flip the last coin and put it at the front (so different from n=3 here). Again, I read no explanation why that would necessarily give you the best probability, but since that is true for all n>3 there is probably something in that. But I'm not sure it's worth beating yourself up for, for limited insight.

singingbanana
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Hey nice vid. I found a slightly different way to solve this problem...I made an 8x8 stochastic transition matrix where each cell is the probability of going to the next.
DId it all in software and got the exact same answer as you did at 7:39, so your work is totally correct :)
God I hope you didn't have to do all of that by hand...but if you did it's amazing you did it perfectly.

CarrotCakeMake
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I did your puzzle while this was rendering. It's full of mistakes I want to change, but I thought it was important to get this one up quickly as it's in response to something topical. 14 slides took blummin' ages.

singingbanana
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Given Player 1 HTH for instance, why are THT and HHT any different to prefer? (I get why it should be something-HT.)

BethanyLowe
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well of course you can work out individually for each choice your opponent makes what the best choice for you is, you could also just write out the eight possibilities for which three tosses come before his and see which choice you make will win the most of them. i was kind of hoping for an explanation of why that strategy is best other than just you can work it out for each possible combination of choices.

ajbcx
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I weighed six oxen and can predict next week lotto balls to be 02 19 20 21 32 and 5000.

singingbanana
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in the case with the W/B balls you can use:
P(AL1W) = 1 - P(B, B) = 1 - 3/5*2/4 = 1 - 3/10 = 7/10
my question is, can you use the prob complement in the coin game to simplify the maths?

scotlandyard
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I think I found a mistake on your posters.

On the poster that's shown at 7:50 (that same poster is also shown in Part 1) the Prob P2 wins column reads:
7/8 3/4 2/3 2/3 7/8 3/4 2/3 2/3

I think you meant it to read:
7/8 3/4 2/3 2/3 2/3 2/3 3/4 7/8

Am I right?

dkusalik
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But i am a little skeptical...Oh uhh did the explanation video came out i cant find it anywhere

Nemesiss