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In this case, there are two values for which f is maximal: (n + 1) p and (n + 1) p − 1. M is the most probable outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the mode. Equivalently, M − p < np ≤ M + 1 − p. Taking the floor function, we obtain M = floor(np). [note 1]
Odds of 4/1 would imply that the bettor stands to make a £400 profit on a £100 stake. If the odds are 1/4, the bettor will make £25 on a £100 stake. In either case, having won, the bettor always receives the original stake back; so if the odds are 4/1 the bettor receives a total of £500 (£400 plus the original £100).
The development of probability theory in the late 1400s was attributed to gambling; when playing a game with high stakes, players wanted to know what the chance of winning would be. In 1494, Fra Luca Paccioli released his work Summa de arithmetica, geometria, proportioni e proportionalita which was the first written text on probability.
1/52! chance of a specific shuffle Mathematics: The chances of shuffling a standard 52-card deck in any specific order is around 1.24 × 10 −68 (or exactly 1 ⁄ 52!) [4] Computing: The number 1.4 × 10 −45 is approximately equal to the smallest positive non-zero value that can be represented by a single-precision IEEE floating-point value.
The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin.
1 in 1 744 278: Every 4776 years (once in recorded history) μ ± 5.5σ: 0.999 999 962 020 875: 3.798 × 10 −8 = 37.98 ppb: 1 in 26 330 254: Every 72 090 years (thrice in history of modern humankind) μ ± 6σ: 0.999 999 998 026 825: 1.973 × 10 −9 = 1.973 ppb: 1 in 506 797 346: Every 1.38 million years (twice in history of humankind) μ ± ...
So there is now a 1 in 48 chance of predicting this number. Thus for each of the 49 ways of choosing the first number there are 48 different ways of choosing the second. This means that the probability of correctly predicting 2 numbers drawn from 49 in the correct order is calculated as 1 in 49 × 48. On drawing the third number there are only ...
For an event X that occurs with very low probability of 0.0000001%, or once in one billion trials, in any single sample (see also almost never), considering 1,000,000,000 as a "truly large" number of independent samples gives the probability of occurrence of X equal to 1 − 0.999999999 1000000000 ≈ 0.63 = 63% and a number of independent ...