If you have a standard, 6-face die, then there are six possible outcomes, namely the numbers from 1 to 6. When you look at all the things that may occur, the formula (just as our coin flip probability formula) states that Classical probability problems often need to you find how often one outcome occurs versus another, and how one event happening affects the probability of future events happening. Therefore, THH should have a higher probability of occurring first.The probability of some event happening is a mathematical (numerical) representation of how likely it is to happen, where a probability of 1 means that an event will always happen, while a probability of 0 means that it will never happen. If I’m wrong on the third flip with HTH, I have to flip again to try to get the first letter H. The first two letters in both sequences have an equal chance of happening, but with the third flip, if I’m wrong with THH, I already have the first letter (T) to begin the sequence again. This leads me to think that odds for HTH vs. This extra flip makes the sequence less likely to occur than not having to have the extra flip to begin the sequence again. With HH, if I’m wrong on the second flip and get a T I have to flip again to try to get the first correct letter to begin the sequence again. TH has a better chance of winning because in the second sequence (TH) if I flip a coin and am wrong (get a T instead of an H) I already have the correct letter to start the sequence again. To explain the reasoning take the HH vs TH example. Figuring out which is which requires some powerful tools.Īt least with two and three coin sequence there is a way to figure out which sequence is better without calculating the odds. So, which do you prefer out of HHHTT and HTTHH? One of them wins 7/13 of the time, and the other 6/13. The algorithm quickly calculates a table of values.Įven a grid of all the 8-flip games can be easily calculated. If just the odds are wanted instead of the sequences, there is an easier way, developed by John Conway. Fortunately, someone pointed out my mistake, and this world of strange games opened up for me as I harnessed the full power of Mathematica. But that was the wrong way to do it, and the odds were all off. I only looked at games up to a certain short length, and then approximated the odds from there. Ī few years ago, I tried to present this problem quickly, and ended up getting the math wrong. The fraction 22/7 is a fairly good approximation for, so the odds of a win are about 1/. One 4-flip game with strange odds is HHHH vs. Every game tends to beat the game in front of it. Notice the clockwise ring of arrows on the outside. HHHH and TTTT are losers, from an odds standpoint. Repeating all of this for the 4-flip games, here are the 14 games with non-obvious strategies. With results from games of 200 flips or less, the odds can be determined very precisely. From that, many more terms were calculated. The sequence from the brute-force code leads to a generating function, which can be used to find many more terms in the sequence.Īgain, the start of a sequence was found by brute-force methods, and that sequence led to an elegant function. With the sequence, FindSequenceFunction or FindGeneratingFunction can be used to get the underlying function. A head and a tail are added to all games without a winner, and then the process is iterated. The set of flips that could have a winner is calculated, and then all the winners are removed. The sequences of a game can be calculated with the piece of code in the downloadable Computable Document Format (CDF) file. The corresponding number of wins is the beginning of the Fibonacci sequence. Here are the ways HHH can win after 3, 4, 5, and 6 flips. For any game your opponent picks, you can pick a game that beats it, making this a nontransitive game. Calculating these odds can be both tedious and mathematically demanding-a natural job for Mathematica or Mathematica Home Edition.Īs a picture, here are the 6 games that are not obvious losers. Here is a table of odds and facts for various 3-flip games. This is the strange world of Penney’s game. Which one? The odds of the event occurring are given at the end. Phrased a different way: Suppose I offer a bet on a series of coin flips. TH, HH will win if the first two flips are HH and will lose if any of those flips are tails. Beforehand, the players each pick a sequence of flips.
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