Variation With Playing Cards
One suggested variation on Penney’s Game uses a pack of ordinary playing cards. The Humble-Nishiyama Randomness Game follows the same format using Red and Black cards, instead of Heads and Tails. At the start of a game each player decides on their three colour sequence for the whole game. Every time the 1st or 2nd player sequence of cards appears, all those cards are removed from the game as a “winning trick” and all cards that have already been turned over are discarded. This continues until the full pack of 52 cards is used. At the end the player with the most “tricks” is declared the winner. An average game will consist of around 7 “tricks”. Due to the repeated nature of this game, the second player`s chance of winning is greatly increased.
Below are the probabilities of the outcomes for each strategy.
| 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 chance of a draw. The odds of the second player winning in such a game 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 |
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