Fair coin

In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin. John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a crown and coated on one side with lead landed heads (wooden side up) 679 times out of 1000. In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the method of the toss, where with sufficient practice a coin can be made to land heads 100% of the time. Exploring the problem of checking whether a coin is fair is a well-established pedagogical tool in teaching statistics. In probability theory, a fair coin is defined as a probability space , which is in turn defined by the sample space, event space, and probability measure. Using for heads and for tails, the sample space of a coin is defined as: The event space for a coin includes all sets of outcomes from the sample space which can be assigned a probability, which is the full power set . Thus, the event space is defined as: is the event where neither outcome happens (which is impossible and can therefore be assigned 0 probability), and is the event where either outcome happens, (which is guaranteed and can be assigned 1 probability). Because the coin is fair, the possibility of any single outcome is 50-50. The probability measure is then defined by the function: So the full probability space which defines a fair coin is the triplet as defined above.