An urn contains n balls - the ith having weight W(i), i=1, …, n . The balls are removed without

step 1 Guys, in this problem, if the ball j is selected before poll number one, exactly we have probabi

An urn contains n balls - the ith having weight W(i), i=1,...

An urn contains $n$ balls - the ith having weight $W(i), i=1, \ldots, n .$ The balls are removed without replacement one at a time according to the following rule: At each selection, the probability that a given ball in the urn is chosen is equal to its weight divided by the sum of the weights remaining in the urn. For instance, if at some time $i_{1}, \ldots, i_{r}$ is the set of balls remaining in the urn, then the next selection will be $i_{j}$ with probability $W\left(i_{j}\right) \mid \sum_{k=1}^{r} W\left(i_{k}\right)$, $j=1, \ldots, r .$ Compute the expected number of balls that are withdrawn before ball number 1 .

An urn contains $n$ balls - the ith having weight $W(i), i=1, \ldots, n .$ The balls are removed without replacement one at a time according to the following rule: At each selection, the probability that a given ball in the urn is chosen is equal to its weight divided by the sum of the weights remaining in the urn. For instance, if at some time $i_{1}, \ldots, i_{r}$ is the set of balls remaining in the urn, then the next selection will be $i_{j}$ with probability $W\left(i_{j}\right) \mid \sum_{k=1}^{r} W\left(i_{k}\right)$,
$j=1, \ldots, r .$ Compute the expected number of balls that are withdrawn before ball number 1 .

Key Concepts

Weighted Sampling Without Replacement

This concept involves selecting items one at a time from a finite set where each item has an associated weight, and once an item is selected, it is not returned to the pool. The probability of selecting an item at any given step is proportional to its weight relative to the sum of weights of the items remaining. This forms the basis for many urn models and is central to analyzing such problems in probability theory.

Sequential Selection Process

In this process, items are removed one by one, and the outcome at each step depends on the composition of the set at that moment. This sequential dependency often requires updating the probabilities at each selection step based on the remaining items, and it is crucial for understanding the dynamics and calculations involved in processes like the one described.

Expected Value

The expected value is a fundamental concept in probability that provides the average outcome of a random process over many repetitions. In the context of sequential removal processes, it can be used to compute the average number of draws or events (such as balls withdrawn before a specific ball) by summing or integrating probabilities, often leveraging linearity of expectation.

Linearity of Expectation

This property states that the expected value of a sum of random variables is equal to the sum of their expected values, regardless of whether the variables are independent. It simplifies the process of calculating overall expected outcomes in complex probabilistic models, such as determining the expected count of events in sequential draw scenarios.

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