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A representation of the possible outcomes of flipping a fair coin four times in terms of the number of heads. As can be seen, the probability of getting exactly two heads in four flips is 6/16 = 3/8, which matches the calculations. For this experiment, let a heads be defined as a success and a tails as a failure.
For example, if x represents a sequence of coin flips, then the associated Bernoulli sequence is the list of natural numbers or time-points for which the coin toss outcome is heads. So defined, a Bernoulli sequence Z x {\displaystyle \mathbb {Z} ^{x}} is also a random subset of the index set, the natural numbers N {\displaystyle \mathbb {N} } .
(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%)
Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to randomly choose between two alternatives. It is a form of sortition which inherently has two possible outcomes.
In probability theory, a tree diagram may be used to represent a probability space. A tree diagram may represent a series of independent events (such as a set of coin flips) or conditional probabilities (such as drawing cards from a deck, without replacing the cards). [1] Each node on the diagram represents an event and is associated with the ...
The first time heads appears, the game ends and the player wins whatever is the current stake. Thus the player wins 2 dollars if heads appears on the first toss, 4 dollars if tails appears on the first toss and heads on the second, 8 dollars if tails appears on the first two tosses and heads on the third, and so on.
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.
For example, when tossing an ordinary coin, one typically assumes that the outcomes "head" and "tail" are equally likely to occur. An implicit assumption that all outcomes are equally likely underpins most randomization tools used in common games of chance (e.g. rolling dice , shuffling cards , spinning tops or wheels, drawing lots , etc.).