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Coins in a fountain is a problem in combinatorial mathematics that involves a generating function.In this problem, a fountain is an arrangement of non-overlapping unit circles into horizontal rows in the plane so that consecutive circles in the bottom row are tangent to each other, and such that each circle in a higher row is tangent to two coins from the next row below it.
It is based on the coin flip used widely in sports and other situations where it is required to give two parties the same chance of winning. Either a specially designed chip or more usually a simple currency coin is used, although the latter might be slightly "unfair" due to an asymmetrical weight distribution, which might cause one state to ...
Odds and evens - a game with the same strategic structure, that is played with fingers instead of coins. Rock paper scissors - a similar game in which each player has three strategies instead of two. Parity game - a much more complicated two-player logic game, played on a colored graph. Penney's game - an exploitable sequence game
As this card-based version is quite similar to multiple repetitions of the original coin game, the second player's advantage is greatly amplified. The probabilities are slightly different because the odds for each flip of a coin are independent while the odds of drawing a red or black card each time is dependent on previous draws. Note that HHT ...
The first die roll selects a row in the table and the second a column. So, for example, a roll of 2 followed by a roll of 4 would select the letter "j" from the fractionation table below. [10] To generate upper/lower case characters or some symbols a coin flip can be used, heads capital, tails lower case.
Another way to generate the same trees is to make a sequence of coin flips, with probability of heads and probability of tails, until the first flip at which the number of tails exceeds the number of heads (for the model in which an external root is allowed) or exceeds one plus the number of heads (when the root must be internal), and then use ...
Entropy Η(X) (i.e. the expected surprisal) of a coin flip, measured in bits, graphed versus the bias of the coin Pr(X = 1), where X = 1 represents a result of heads. [ 10 ] : 14–15 Here, the entropy is at most 1 bit, and to communicate the outcome of a coin flip (2 possible values) will require an average of at most 1 bit (exactly 1 bit for ...
The binomial distribution generalizes this to the number of heads from performing n independent flips (Bernoulli trials) of the same coin. The multinomial distribution models the outcome of n experiments, where the outcome of each trial has a categorical distribution , such as rolling a k -sided die n times.