A possible sequence in Penney ‘s game : heads, tails, heads

**Penney’s game**, named after its inventor Walter Penney, is a binary ( head/tail ) sequence generating game between two players. Player A selects a sequence of heads and tails ( of length 3 or larger ), and shows this sequence to player B. Player B then selects another sequence of heads and tails of the same length. subsequently, a fair coin is tossed until either player A ‘s or actor B ‘s sequence appears as a consecutive sequel of the mint pass outcomes. The musician whose sequence appears first wins. Provided sequences of at least distance three are used, the second player ( B ) has an edge over the starting musician ( A ). This is because the game is nontransitive such that for any given sequence of length three or longer one can find another sequence that has higher probability of occurring first.

Reading: Penney’s game – Wikipedia

## psychoanalysis of the three-bit plot [edit ]

For the three- sting sequence game, the moment actor can optimize their odds by choosing sequences according to :

1st player’s choice | 2nd player’s choice | Odds in favour of 2nd player |
---|---|---|

HHH |
THH |
7 to 1 |

HHT |
THH |
3 to 1 |

HTH |
HHT |
2 to 1 |

HTT |
HHT |
2 to 1 |

THH |
TTH |
2 to 1 |

THT |
TTH |
2 to 1 |

TTH |
HTT |
3 to 1 |

TTT |
HTT |
7 to 1 |

An easy way to remember the sequence is for the irregular player to start with the inverse of the middle choice of the first player, then follow it with the first player ‘s first two choices .

- So for the first player’s choice of
**1-2-3** - the second player must choose
**(not-2)-1-2**

where ( not-2 ) is the face-to-face of the moment choice of the first musician. An intuitive explanation for this result is that in any case that the sequence is not immediately the first actor ‘s option, the chances for the first musician getting their sequence-beginning, the open two choices, are normally the casual that the second gear player will be getting their full sequence. So the second gear actor will most likely “ ending before ” the first player.

Read more: All About Australian Coins

## scheme for more than three bits [edit ]

The optimum strategy for the first player ( for any length of the sequence no less than 4 ) was found by J.A. Csirik ( See References ). It is to choose HTTTT … ..TTTHH ( k − 3 { \displaystyle k-3 } T ‘s ) in which case the second player ‘s maximal odds of winning is ( 2 kelvin − 1 + 1 ) : ( 2 potassium − 2 + 1 ) { \displaystyle ( 2^ { k-1 } +1 ) : ( 2^ { k-2 } +1 ) } .

## variation with play cards [edit ]

One suggested pas seul on Penney ‘s Game uses a pack of ordinary play cards. The Humble-Nishiyama Randomness Game follows the same format using Red and Black cards, alternatively of Heads and Tails. [ 1 ] [ 2 ] The crippled is played as follows. At the begin of a game each player decides on their three colour sequence for the wholly game. The cards are then turned over one at a time and placed in a lineage, until one of the choose triples appears. The winning player takes the retrousse cards, having won that “ trick ”. The plot continues with the lie of the unused cards, with players collecting tricks as their triples come up, until all the cards in the pack have been used. The achiever of the game is the player that has won the most tricks. An average bet on will consist of around 7 “ tricks ”. As this card-based version is quite like to multiple repetitions of the original coin game, the second base actor ‘s advantage is greatly amplified. The probabilities are slightly different because the odds for each flip of a mint are independent while the odds of drawing a loss or bootleg circuit board each time is dependent on previous draws. note that HHT is a 2:1 favored over HTH and HTT but the odds are different for BBR over BRB and BRR. Below are approximate probabilities of the outcomes for each strategy based on calculator simulations : [ 3 ]

1st player’s choice | 2nd player’s choice | Probability 1st player wins | Probability 2nd player wins | Probability of a draw |
---|---|---|---|---|

BBB |
RBB |
0.11% | 99.49% | 0.40% |

BBR |
RBB |
2.62% | 93.54% | 3.84% |

BRB |
BBR |
11.61% | 80.11% | 8.28% |

BRR |
BBR |
5.18% | 88.29% | 6.53% |

RBB |
RRB |
5.18% | 88.29% | 6.53% |

RBR |
RRB |
11.61% | 80.11% | 8.28% |

RRB |
BRR |
2.62% | 93.54% | 3.84% |

RRR |
BRR |
0.11% | 99.49% | 0.40% |

If the game is ended after the first trick, there is a negligible find of a draw. The odds of the second player winning in such a bet on appear in the table below .

1st player’s choice | 2nd player’s choice | Odds in favour of 2nd player |
---|---|---|

BBB |
RBB |
7.50 to 1 |

BBR |
RBB |
3.08 to 1 |

BRB |
BBR |
1.99 to 1 |

BRR |
BBR |
2.04 to 1 |

RBB |
RRB |
2.04 to 1 |

RBR |
RRB |
1.99 to 1 |

RRB |
BRR |
3.08 to 1 |

RRR |
BRR |
7.50 to 1 |

## version with a Roulette bicycle [edit ]

recently Robert W. Vallin, and former Vallin and Aaron M. Montgomery, presented results with Penney ‘s Game as it applies to ( American ) roulette with Players choosing Red/Black rather than Heads/Tails. In this situation the probability of the ball landing on crimson or black is 9/19 and the remaining 1/19 is the find the musket ball lands on green for the numbers 0 and 00. There are respective ways to interpret k : ( 1 ) as a “ raving mad circuit board ” so that BGR can be read at Black, Black, Red and Black, Red, Red, ( 2 ) as a do-over, the game stops when greens appears and restarts with the following spin, ( 3 ) as merely itself with not extra rendition. Results have been worked out for odds and wait times. [ 4 ]

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