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Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p.Three examples are shown: Blue curve: Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 0.
The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables and the conditional probability distributions, which deal with how the outputs of one random variable are distributed when ...
However, because of the widespread use of random variables, which transform the sample space into a set of numbers (e.g., , ), it is more common to study probability distributions whose argument are subsets of these particular kinds of sets (number sets), [6] and all probability distributions discussed in this article are of this type.
An illustration of the convergence of sequence averages of rolls of a dice to the expected value of 3.5 as the number of rolls (trials) grows. Let represent the outcome of a roll of a fair six-sided die. More specifically, will be the number of pips showing on the top
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Thus, the probability that a number starts with the digits 3, 1, 4 (some examples are 3.14, 3.142, π, 314280.7, and 0.00314005) is log 10 (1 + 1/314) ≈ 0.00138, as in the box with the log-log graph on the right. This result can be used to find the probability that a particular digit occurs at a given position within a number.
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