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Analyzing Random Ball Assignments to Bins: Maximum Capacity and Expectation Insights

This article explores the fascinating problem of randomly assigning balls to bins. Specifically, it addresses two primary questions: What is the maximum number of balls that can end up in any single bin? And what is the expected number of bins containing a specified number of balls? Through the application of probabilistic methods, including upper bounds on binomial coefficients and inequalities like Markov and Chebyshev, we derive insights that can be applied in various scenarios, including variations of the famous Birthday Problem. We aim to elucidate the behavior of random distributions in bin systems.

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Analyzing Random Ball Assignments to Bins: Maximum Capacity and Expectation Insights

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Presentation Transcript

  1. Occupancy Problems m balls being randomly assigned to one of n bins. (Independently and uniformly) The questions: - what is the maximum number of balls in any bin? -what is the expected number of bins with k balls?

  2. For arbitrary events: , not necessarily independent: the probability of the union of events is no more than the sum of their probabilities. Let m=n : For let , where j is the number of balls in the th bin. Then we get: for all i.

  3. Now: we concentrate on analyzing the 1st bin, so: Let denote the event that bin has or more balls in it. So: From upper bound for binomial coefficients

  4. Now, let Then: with probability at least , no bin has more than balls in it!

  5. The Birthday Problem Now n=365, How large must m be before two people in the group are likely to share their birthday? For , let denote the event that the th ball lands in a bin not containing any of the first balls. But:

  6. So: Now we can see that for the probability that all m balls land in distinct bins is at most .

  7. Markov Inequality Let Y be a random variable assuming only non-negative values. Than for all : Or: Proof:define Than: Now, , , else

  8. standard deviation If X is a random variable with expectation , The variance is defined: The standard deviation of X is

  9. Chebyshev’s Inequality Let X be a random variable with expectation , and standard deviation . Then for any : Proof: First: for we get . applying the markov inequality to Y bounds this probability.

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