Consider n independent trials, each of which results in one...

Consider $n$ independent trials, each of which results in one of the outcomes $1, \ldots, k$ with respective probabilities $p_{1}, \ldots, p_{k}, \quad \sum_{i=1}^{k} p_{i}=1 .$ Show that if all the $p_{i}$ are small, then the probability that no trial outcome occurs more than once is approximately equal to $\exp (-n(n-1)$ $\left.\sum_{i} p_{i}^{2} / 2\right)$.

Key Concepts

Independent Trials

This concept involves considering each trial as having no influence on the outcomes of the other trials. Independence allows us to multiply individual probabilities to compute the joint probability of events occurring across trials, making it a fundamental assumption in calculating the probability that no individual outcome is repeated across the set of trials.

Combinatorial Counting

In problems of repeated trials, combinatorial counting is used to enumerate the number of potential pairs (or groups) of trials, particularly when considering events such as collisions—where the same outcome appears more than once. This counts the number of ways in which pairwise comparisons can take place, which is crucial for estimating the probability of any repetition across the trials.

Small Probability Approximations

When the individual outcome probabilities are small, higher order terms become negligible, and approximations such as 1 - x ? exp(-x) for small x become valid. This allows simplification of the probabilities involving repeated outcomes and facilitates a more tractable analysis by substituting the product of probabilities with an exponential function.

Exponential Approximation

The exponential approximation emerges from the defined limit behavior of the binomial or multinomial expressions under rare event conditions. By approximating the probability that no outcome is repeated by an exponential function, it becomes easier to handle the analysis mathematically, especially in large-scale problems. This method is analogous to the Poisson process where events occur independently and infrequently.

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