bramp.net :: Derren Brown’s Coin Game

Derren Brown’s Coin Game

2009-09-12 01:09:13 5 Comments

Update: I found out this coin game is called Penney's Game and is played slightly differently to how I thought.

On TV tonight Derren Brown (poorly) described a game where two players would each pick a sequence of 3 heads or tails and then keep flipping a coin until they obtained their sequence. The winner would be the one that flips their own sequence the most times. Derren suggested that by having a team of supporters they could influence the coin tosses and allow you to win. In the programme the player with the supports won, but how?

So to start with the basics there are 8 different sequences to pick:

Combination	Probability
HHH	1/8
HHT	1/8
HTH	1/8
HTT	1/8
THH	1/8
THT	1/8
TTH	1/8
TTT	1/8

Table of probabilities for three coin flip combinations

Each sequence of flips is equally likely, and on face value you would expect a 1 in 8 chance of any of these combinations to occur. Now this is true, however, if you play the game in the same way that Derren Brown does, then a different outcome will happen.

The difference in Derren’s game is what happens when you fail to flip the correct face. Instead of starting from the beginning you are allowed to carry on. So instead of having to get 3 in a row from scratch, you have to get your sequence from the last 3 attempts. The player would score a point with the combination HTT if they had previous flipped HTHTT.

This can be used to your advance by picking a sequence which starts and ends with different sides. If, for example, your sequence starts with a head and ends in a tail, then if you fail to flip the 3rd coin correctly, then you have already successfully flipped once for the next attempt.

The worst combinations to pick are HHH and TTT because you can never capitalise on a previous bad flip. The next worst are HTH or THT, as with these you can capitalise on one previous flip if you fail to flip the 2nd coin. Finally the best combinations are, HHT, HTT, THH, TTH, as here you can always capitalise on a bad flip regardless if it is the second or third flip.

Because I’m too lazy to work out the exact odds, I used Monte-Carlo simulations to generate this table of results:

Combination	Percentage Outcome
HHH	8.4697%
HHT	14.8300%
HTH	11.8627%
HTT	14.8335%
THH	14.8326%
THT	11.8603%
TTH	14.8356%
TTT	8.4756%

Table of probabilities for finding three coin flips in a sequence

Derren suggested that if you ever played this game you should let your opponent chose first, and then you should transpose their sequence by flipping the middle value, placing it on the front, and then dropping the fourth value. So, for example, HHH has its middle value of H, flip this and move it to the front to make THHH, and finally drop the final H, leaving us with THH. If you do this with any combination you always get one of the better combinations. However, Derren incorrectly explained that this combination would always win over your opponent, this is not true if they picked one of the four best combinations. I did notice however, that this is true if you are playing with 4 flip sequences, and I even found a Wolfram demonstration of this.

Kate

Thank you for the explanation of why some sequences are better than others to choose, plus data too! But I don't fully understand how it explains Derren Brown's algorithm, which is to take your opponent's sequence, flip the middle toss and place it at the start, and drop the end toss. By this method, Brown claims that if the first person picked, say, HHT, then the second person will always beat them by picking THH, and yet your chart shows that HTT and THH have the same probability. I think there's more to it...

2009-09-16 18:54:20

OK, following on from my own comment here, I have the Derren Brown trick figured out (I think). The second player *will* win, and this is why:

Imagine a long sequence of heads and tails, randomly generated. Essentially what you do is read off each triplet, then slide the triplet frame, as it were, along by one, and read again. If a triplet matched one or other player's choices, the next triplet, and the one after, don't get counted and therefore can't generate wins (because the first one or two members of that triplet belonged to the previous winning sequence, and are locked out, as it were, by the fact that frame-reading re-starts from the beginning after a win). Thus, a win deprives the next two triplets of the ability to contribute to the total score. Now, because of the rule whereby the second player flips the middle member of the first player's triplet choice, the next triplet after a player 1 win, and which gets blocked by that win, is never a player 2 sequence. Why? Because that triplet starts with the middle member of the previously winning sequence, and the flipping rule makes it therefore *not* the start of the player 2 sequence. Thus, the triplet that got blocked by a win was never going to be a player 2 sequence, but might still be a player 1 sequence. Player 2 is slightly protected, as it were, from the lockout phenomenon, whereas player 1 isn't and therefore preferentially accumulates blocked triplets that get left out of the final count.

Does that makes sense?!

2009-09-17 11:28:37

bramp

Hey Kate,
Glad you found and enjoyed my write up.

I think what you have said does make sense, however, I think that means only one person is flipping coins, instead of each player having their own sequence of heads and tails. From what I remember from last week's show each player was flipping their own sequence.

I did find a link on wolfram.com which played this game with a single sequence of flips (like you suggest) and with quadruplet instead of triplets. In that game each combination was always beaten by another.

I look forward to tonight's episode, I hope it doesn't disappoint me :(

2009-09-18 17:40:55

I think there was only one sequence... otherwise they wouldn't interact. If there is only one sequence then Brown's trick works, but only partly for the reason I thought. In fact the main reason (I tried this out, yes OK I got a bit obsessed!) is the overlap between the first player's first two choices and the second player's second two. Because the last part of player 2's sequence overlaps the beginning of player 1, for 50% of times when player 1 completed their sequence, player 2 will have completed theirs one toss before, thus blocking player 1's win. The only time this doesn't work is if player 1 chooses a palindromic sequence like HTH. In this case, if you don't flip the middle toss and you choose THT, both sequences overlap each other and so both wins block each other. The flipping rule stops player 1 from being able to block player 2.

Last night's episode was a bit disturbing... it amazes and depresses me that people can be so suggestible!

2009-09-19 19:37:34

Last night's episode was disappointing, I've yet to find a single person that was actually stuck to their chair :(

2009-09-19 23:02:34