The desired useful approximation is given by the central limit theorem, which in the special case of the binomial distribution was first discovered by Abraham de Moivre about 1730. Let X₁,…, X_n be independent random variables having a common distribution with expectation μ and variance σ². The law of large numbers implies that the distribution of the random variable X̄_n = n⁻¹(X₁ +⋯+ X_n) is essentially just the degenerate distribution of the constant μ, because E(X̄_n) = μ and Var(X̄_n) = σ²/n → 0 as n → ∞. The standardized random variable (X̄_n − μ)/(σ/Square root of√n) has mean 0 and variance 1. The central limit theorem gives the remarkable result that, for any real numbers a and b, as n → ∞,where

Thus, if n is large, the standardized average has a distribution that is approximately the same, regardless of the original distribution of the Xs. The equation also illustrates clearly the square root law: the accuracy of X̄_n as an estimator of μ is inversely proportional to the square root of the sample size n.

Use of equation (12) to evaluate approximately the probability on the left-hand side of equation (11), by setting b = −a = εSquare root of√n/σ, yields the approximation G(εSquare root of√n/σ) − G(−εSquare root of√n/σ). Since G(2) − G(−2) is approximately 0.95, n must be about 4σ²/ε² in order that the difference |X̄_n − μ| will be less than ε with probability 0.95. For the special case of the binomial distribution, one can again use the inequality σ² = p(1 − p) ≤ 1/4 and now conclude that about 1,100 balls must be drawn from the urn in order that the empirical proportion of red balls drawn will be within 0.03 of the true proportion of red balls with probability about 0.95. The frequently appearing statement in newspapers in the United States that a given opinion poll involving a sample of about 1,100 persons has a sampling error of no more than 3 percent is based on this kind of calculation. The qualification that this 3 percent sampling error may be exceeded in about 5 percent of the cases is often omitted. (The actual situation in opinion polls or sample surveys generally is more complicated. The sample is drawn without replacement, so, strictly speaking, the binomial distribution is not applicable. However, the “urn”—i.e., the population from which the sample is drawn—is extremely large, in many cases infinitely large for practical purposes. Hence, the composition of the urn is effectively the same throughout the sampling process, and the binomial distribution applies as an approximation. Also, the population is usually stratified into relatively homogeneous groups, and the survey is designed to take advantage of this stratification. To pursue the analogy with urn models, one can imagine the balls to be in several urns in varying proportions, and one must decide how to allocate the n draws from the various urns so as to estimate efficiently the overall proportion of red balls.)

Considerable effort has been put into generalizing both the law of large numbers and the central limit theorem, so that it is unnecessary for the variables to be either independent or identically distributed.

The law of large numbers discussed above is often called the “weak law of large numbers,” to distinguish it from the “strong law,” a conceptually different result discussed below in the section on infinite probability spaces.

The Poisson approximation

The weak law of large numbers and the central limit theorem give information about the distribution of the proportion of successes in a large number of independent trials when the probability of success on each trial is p. In the mathematical formulation of these results, it is assumed that p is an arbitrary, but fixed, number in the interval (0, 1) and n → ∞, so that the expected number of successes in the n trials, np, also increases toward +∞ with n. A rather different kind of approximation is of interest when n is large and the probability p of success on a single trial is inversely proportional to n, so that np = μ is a fixed number even though n → ∞. An example is the following simple model of radioactive decay of a source consisting of a large number of atoms, which independently of one another decay by spontaneously emitting a particle. The time scale is divided into a large number of very small intervals of equal lengths, and in each interval, independently of what happens in the other intervals, the source emits one or no particle with probability p or q = 1 − p respectively. It is assumed that the intervals are so small that the probability of two or more particles being emitted in a single interval is negligible. One now imagines that the size of the intervals shrinks to 0, so that the number of trials up to any fixed time t becomes infinite. It is reasonable to assume that the probability of emission during a short time interval is proportional to the length of the interval. The result is a different kind of approximation to the binomial distribution, called the Poisson distribution (after the French mathematician Siméon-Denis Poisson) or the law of small numbers.

Assume, then, that a biased coin having probability p = μδ of heads is tossed once in each time interval of length δ, so that by time t the total number of tosses is an integer n approximately equal to t/δ. Introducing these values into the binomial equation and passing to the limit as δ → 0 gives as the distribution for N(t) the number of radioactive particles emitted in time t:

The right-hand side of this equation is the Poisson distribution. Its mean and variance are both equal to μt. Although the Poisson approximation is not comparable to the central limit theorem in importance, it nevertheless provides one of the basic building blocks in the theory of stochastic processes.

Infinite sample spaces

The experiments described in the preceding discussion involve finite sample spaces for the most part, although the central limit theorem and the Poisson approximation involve limiting operations and hence lead to integrals and infinite series. In a finite sample space, calculation of the probability of an event A is conceptually straightforward because the principle of additivity tells one to calculate the probability of a complicated event as the sum of the probabilities of the individual experimental outcomes whose union defines the event.

Experiments having a continuum of possible outcomes—for example, that of selecting a number at random from the interval [r, s]—involve subtle mathematical difficulties that were not satisfactorily resolved until the 20th century. If one chooses a number at random from [r, s], the probability that the number falls in any interval [x, y] must be proportional to the length of that interval; and, since the probability of the entire sample space [r, s] equals 1, the constant of proportionality equals 1/(s − r). Hence, the probability of obtaining a number in the interval [x, y] equals (y − x)/(s − r). From this and the principle of additivity one can determine the probability of any event that can be expressed as a finite union of intervals. There are, however, very complicated sets having no simple relation to the intervals—e.g., the rational numbers—and it is not immediately clear what the probabilities of these sets should be. Also, the probability of selecting exactly the number x must be 0, because the set consisting of x alone is contained in the interval [x, x + 1/n] for all n and hence must have no larger probability than 1/[n(s − r)], no matter how large n is. Consequently, it makes no sense to try to compute the probability of an event by “adding” the probabilities of the individual outcomes making up the event, because each individual outcome has probability 0.

A closely related experiment, although at first there appears to be no connection, arises as follows. Suppose that a coin is tossed n times, and let X_k = 1 or 0 according as the outcome of the kth toss is heads or tails. The weak law of large numbers given above says that a certain sequence of numbers—namely the sequence of probabilities given in equation (11) and defined in terms of these n Xs—converges to 1 as n → ∞. In order to formulate this result, it is only necessary to imagine that one can toss the coin n times and that this finite number of tosses can be arbitrarily large. In other words, there is a sequence of experiments, but each one involves a finite sample space. It is also natural to ask whether the sequence of random variables (X₁ +⋯+ X_n)/n converges as n → ∞. However, this question cannot even be formulated mathematically unless infinitely many Xs can be defined on the same sample space, which in turn requires that the underlying experiment involve an actual infinity of coin tosses.

For the conceptual experiment of tossing a fair coin infinitely many times, the sequence of zeros and ones, (X₁, X₂,…), can be identified with that real number that has the Xs as the coefficients of its expansion in the base 2, namely X₁/2¹ + X₂/2² + X₃/2³ +⋯. For example, the outcome of getting heads on the first two tosses and tails thereafter corresponds to the real number 1/2 + 1/4 + 0/8 +⋯ = 3/4. (There are some technical mathematical difficulties that arise from the fact that some numbers have two representations. Obviously 1/2 = 1/2 + 0/4 +⋯, and the formula for the sum of an infinite geometric series shows that it also equals 0/2 + 1/4 + 1/8 +⋯. It can be shown that these difficulties do not pose a serious problem, and they are ignored in the subsequent discussion.) For any particular specification i₁, i₂,…, i_n of zeros and ones, the event {X₁ = i₁, X₂ = i₂,…, X_n = i_n} must have probability 1/2ⁿ in order to be consistent with the experiment of tossing the coin only n times. Moreover, this event corresponds to the interval of real numbers [i₁/2¹ + i₂/2² +⋯+ i_n/2ⁿ, i₁/2¹ + i₂/2² +⋯+ i_n/2ⁿ + 1/2ⁿ] of length 1/2ⁿ, since any continuation X_{n + 1}, X_{n + 2},… corresponds to a number that is at least 0 and at most 1/2^{n + 1} + 1/2^{n + 2} +⋯ = 1/2ⁿ by the formula for an infinite geometric series. It follows that the mathematical model for choosing a number at random from [0, 1] and that of tossing a fair coin infinitely many times assign the same probabilities to all intervals of the form [k/2ⁿ, 1/2ⁿ].