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The likelihood that a given event will occur expressed as the ratio of the number of actual occurrences, n, to the number of possible occurrences, N: n/N; where all of N are equally likely. For example, when throwing a die there is 1 way in which a six can turn up and 5 ways in which a "not six" can occur. Thus n = 1 and N = 5 + 1 = 6, and the ratio n/N = 1/6. If two dice are thrown there are 6 × 6 (= 36) possible pairs of numbers that can turn up: the chance of throwing two sixes is 1/36. This does not mean that if a six has just been thrown there is only a 1/36 chance of throwing another: the two events are independent; the probability of their occurring together is 1/36.

Consider throwing a die six times with the aim of getting each of the numbers exactly once. If you want to do this in order (a permutation), say from 1 to 6, working out the probability of your doing so is easy: you have 1/6 chance of throwing a one, 1/6 chance of throwing a two, and so on; so that the probability of a favorable result overall is (1/6)6 = 1/46656. If you are not concerned with the order (combination) the situation is different: the probability of a favorable result on the first throw is 1 (any number is favorable), on the second 5/6, and so on, so the overall probability is
6/6 × 5/6 × 4/6 × 3/6 × 2/6 × 1/66
or 6!/66 (see factorial) – 1/65. Clearly one is more likely to succeed with a desired combination than with a permutation.

Probability theory is plainly intimately linked with statistics. More advanced probability theory has contributed vital understandings in many fields of physics, as in thermodynamics, behavior of particles in a colloid (see Brownian motion, or molecules in a gas, and atomic physics.

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