A total of n balls, numbered 1 through n, are put into n...

A total of $n$ balls, numbered 1 through $n$, are put into $n$ urns, also numbered 1 through $n$ in such a way that ball $i$ is equally likely to go into any of the ums $1,2, \ldots, i$. Find (a) the expected number of urns that are empty; (b) the probability that none of the urns is empty.

A total of $n$ balls, numbered 1 through $n$, are put into $n$ urns, also numbered 1 through $n$ in such a way that ball $i$ is equally likely to go into any of the ums $1,2, \ldots, i$. Find
(a) the expected number of urns that are empty;
(b) the probability that none of the urns is empty.

Key Concepts

Indicator Random Variables

Indicator random variables are useful tools in probability that take the value 1 if a particular event occurs and 0 otherwise. In problems like these, they help in isolating the event of an urn being empty so that the expected number of empty urns can be computed as the sum of the expectations of these individual indicators.

Linearity of Expectation

Linearity of expectation is a fundamental concept stating that the expected value of a sum of random variables is the sum of their expected values, regardless of whether the variables are dependent. This principle simplifies the computation of total expected counts, such as the total number of empty bins, by allowing one to consider each component separately.

Telescoping Products

Telescoping products involve a series of multiplicative terms that cancel out sequentially, reducing a complicated product to a much simpler expression. This approach is particularly useful for deriving compact closed-form expressions in probability calculations that involve sequential or conditional events.

Permutations and Surjective Mappings

Permutations refer to the different arrangements of a set of objects, whereas surjective mappings are functions that cover every element in the target set. In combinatorial probability, when each container (or urn) is required to be filled exactly once, the problem reduces to counting the number of valid permutations or surjective assignments under given constraints.

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