Suppose that n balls are thrown independently and uniformly...

Suppose that $n$ balls are thrown independently and uniformly at random into $n$ bins. (a) Find the conditional probability that bin 1 has one ball given that exactly one ball fell into the first three bins. (b) Find the conditional expectation of the number of balls in bin 1 under the condition that bin 2 received no balls. (c) Write an expression for the probability that bin 1 receives more balls than bin $2 .$

Suppose that $n$ balls are thrown independently and uniformly at random into $n$ bins.
(a) Find the conditional probability that bin 1 has one ball given that exactly one ball fell into the first three bins.
(b) Find the conditional expectation of the number of balls in bin 1 under the condition that bin 2 received no balls.
(c) Write an expression for the probability that bin 1 receives more balls than bin $2 .$

Key Concepts

Conditional Expectation

This concept extends the idea of expectation (the average outcome) to include additional information or conditions. In the problem, computing the expected number of balls in one bin given that another bin received no balls shows how additional conditions can affect the expected value of a random variable.

Conditional Probability

Conditional probability is used here to compute probabilities given extra information about the outcome. For example, finding the probability that a specific bin receives exactly one ball under the condition that a group of bins received exactly one ball overall involves re?scaling the original probability by the likelihood of the condition being met.

Multinomial Distribution

When multiple balls are allocated into multiple bins, the joint distribution of the number of balls in each bin can be modeled using the multinomial distribution. This distribution is fundamental for calculating probabilities and expectations when dealing with several categories (bins) simultaneously under a uniform assignment mechanism.

Uniform Distribution

In this context, each ball is equally likely to go into any bin, meaning that the probability of any ball landing in any given bin is the same. This uniformity is key to deriving probabilities and expectations, as it ensures identically distributed outcomes for each ball.

Balls and Bins Model

This is a classic probabilistic model where a number of balls are independently and uniformly distributed into bins. It serves as an example of an occupancy problem and is foundational in understanding distributional properties, such as how often a particular bin might receive a certain number of balls.

Independence

The balls are thrown independently, meaning the outcome of one throw does not affect the outcome of any other. This independence simplifies the analysis as it allows us to multiply probabilities and use well?known probability distributions, such as the multinomial distribution, without having to consider interactions between the balls.

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