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This probability is commonly called the Bernoulli measure. [ 2 ] Note that the probability of any specific, infinitely long sequence of coin flips is exactly zero; this is because lim n → ∞ p n = 0 {\displaystyle \lim _{n\to \infty }p^{n}=0} , for any 0 ≤ p < 1 {\displaystyle 0\leq p<1} .
It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have /
The entropy of the unknown result of the next toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers one full bit of information.
(Note: r is the probability of obtaining heads when tossing the same coin once.) Plot of the probability density f(r | H = 7, T = 3) = 1320 r 7 (1 − r) 3 with r ranging from 0 to 1. The probability for an unbiased coin (defined for this purpose as one whose probability of coming down heads is somewhere between 45% and 55%)
Identically distributed: Regardless of whether the coin is fair (with a probability of 1/2 for heads) or biased, as long as the same coin is used for each flip, the probability of getting heads remains consistent across all flips. Such a sequence of i.i.d. variables is also called a Bernoulli process.
A fair coin, when tossed, should have an equal chance of landing either side up. 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.
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...
The two remaining possibilities are equally likely. So the probability that the box is GG, and the other coin is also gold, is 1/2. The reasoning for the 2/3 is as follows: Originally, all six coins were equally likely to be chosen. The chosen coin cannot be from drawer S of box GS, or from either drawer of box SS.