# Tagged Questions

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### Probability that the first $2$ balls are white, given that the sample contains exactly $6$ white balls

An urn contains $30$ white and $15$ black balls. If $10$ balls are drawn without replacement, find the probability that the first $2$ balls are white, given that the sample contains exactly $6$ ...
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### If I randomly throw 10,000 balls to 4,000 buckets, what is the probability that at least one bucket contains more than 10 balls?

If I randomly throw 10,000 balls to 4,000 buckets, what is the probability that at least one bucket contains more than 10 balls? I don't know how to even approach this. A rough number is also ...
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### Probability of assigning balls into buckets, where each bucket has a certain capacity.

I'll start with a specific example of what I am trying to solve: I have eight balls to be randomly placed into four buckets. Buckets #1-3 have the capacity of 2, 2, 3 respectively, while bucket #4 ...
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### Probability of drawing a run of a specific color from an urn with two colors of balls

I was sent a puzzle involving an urn with 128 white balls and 288 black. If the balls are drawn without replacement until the urn is exhausted, what is the probability that a sequence of 10 or more ...
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### Expected number of bins with more than one ball

Suppose that $n$ balls are randomly thrown into $N$ bins. We can compute the expected number of bins that contain at least one ball as $E(X) = N(1 - (1 - 1/N)^n)$. Now, suppose that instead we are ...
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### Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?

I have a very simple simulation program, the sequence is: Create an array of 400k elements Use a PRNG to pick an index, and mark the element (repeat 400k times) Count number of marked elements. An ...
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### Is there a fast, reasonably accurate estimator for multinomial PDF?

I am working on a balls in boxes kind of problem, where the probability of a ball ending up in a certain box varies by box, that is, each box has some probability P of getting any ball, all together ...
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### We throwing $m$ balls to $n$ cells…

We throwing $m$ balls to $n$ cells randomly... At each cell can be more then one ball, or (of course) it can still empty. What is the expectation of the empty cells? I'd like to get any help! Thank ...
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### Find the probability that the $4$th ball removed from the box is white

A box has $10$ balls, $6$ of which are black and $4$ of which are white. $3$ balls are removed from the box, their color unnoted. Find the probability that a fourth ball removed from the box is ...
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### Expected total number of balls in all bins after throwing balls uniformly randomly to bins that have limited capacity

Consider throwing $n$ balls uniformly randomly to $L$ bins. Each bin has capacity $G$, meaning that if a ball is threw to a bin that already has $G$ balls in it, the ball is discarded. Is that ...
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### Expectation time for at least 2 balls in each bin

Given $N$ bins and an unlimited number of balls, to be assigned one ball per time unit to a uniformly random bin. What is the expected time to achieve at least 2 balls in every bin? From a previous ...
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### $N$ bins, $m$ balls: Probability of any bin containing *exactly* $k$ balls.

This is an extension to question 821984: Given $N$ balls independently distributed randomly among $m$ bins, what is the probability that at least one bin gets exactly $k$ balls? And is this ...
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### What is the probability of picking $4$ balls out from the two bins?

Assume there are $2$ bins (bin $a$ and bin $b$ ), and each bin contains $n$ indistinguishable balls with $n>10$. The probability of picking $k$ ball from each bin is $p_k$, ($k\ge 0$). Then what is ...
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### probability that given n balls and 60 urns no urn has more than 1 ball

A carnival game is set up so that a ball put into play has an equal chance of landing in any of 60 different slots. The operator of the game allows you to choose any number of balls and put them all ...
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### Balls and bins question with nlogn balls and n bins.

The question: Suppose we randomly drop nlogn balls into n bins. Give an upper bound on the expectation of the maximum number of balls in any bin. How would this be done? I believe the answer is well ...
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### Balls and bins problem - expected number of balls needed to throw [closed]

Suppose we have n boxes and we start randomly and independently throwing balls into the boxes. (a) For a given box, what is the expected number of balls we need to throw before one of the balls lands ...
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### Ball and bins & Coupon collector problem

If you know a coupon collector problem, you will know what I am talking about. But if you are not familiar with I will try to explain what is the coupon collector problem. I have $n$ bins. I throw ...
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### Bins and balls model - filling first bins [close]

We have $n$ bins and $m$ balls. I want to compute the probability that in the first $k$ bins, $q$ of them will be non-empty. I can throw $m$ balls into $n$ bins in $n^m$ ways. Using Stirling ...
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### Probability and conditional?

A bag contains four red balls and four white balls. A ball is selected at random, removed and replaced by a ball of the opposite colour. A second ball is then selected at random. Calculate the ...
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### Probability that $\frac{n}{2}$ bins are empty [close]

A Bloom filter of length $n$ was built. I have only the first $\frac{n}{2}$ bits of this filter. How will the false positive probability change? For the whole Bloom filter, the false positive ...
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### probability of specific number of balls in an urn

I have $n$ balls, I distribute them between 2 urns. Each ball goes to urn 1 with probability $p$ and urn 2 with probability $1-p$. What is the probability that there are $l$ balls in urn 1 when all ...
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### Expectation of throwing $n$ balls into $n$ bins

Suppose we throw $n$ indistinguishable balls in $n$ bins at random. The throws are independent. What is the expected number of empty bins? What is the expected number of bins with one ball. Using ...
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### Balls and bins probability problem

$k = \sqrt n$ balls are thrown into $n$ bins. The bins are standing in a row and numbered from 1 to $n$. What is the probability that there are no two balls in the same bin or in adjacent bins??? In ...
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### A bin and balls problem

Throw 7 balls into 7 bins. Given there are exactly 2 empty boxes, find the probability that 1 bin contains 3 balls, and thus the other 4 bins contain 1 ball each. I know I could use a multinomial ...
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### Bounds for max load with $n$ balls in $n$ bins?

If you throw $n$ balls into $n$ bins uniformly and independently at random, let $X$ be the number of balls in the bin with the largest number of balls in it. Is there a nice elementary way to ...
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### Expected max load with $n$ balls in $n$ bins?

If you throw $n$ balls into $n$ bins uniformly and independently at random, let $X$ be the number of balls in the bin with the largest number of balls in it. Is there an elementary way to ...
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### Balls and bins conditioned on the number of non-empty bins

The expected number of occupied bins in the standard balls and bins problem (with $m$ balls into $N$ bins) is $N\left( {1 - {{\left( {1 - \frac{1}{N}} \right)}^m}} \right)$. How do I compute this ...
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### Average number of bins occupied when throwing $n$ balls into $N$ bins

There are $n$ balls and $N$ bins. At each time, a ball is thrown in one bin of $N$ bins at random. This repeats n times. So that in total $n$ balls are thrown into bins. The question is, on average, ...
I'm trying to find the probability of an outcome where, using the traditional example, white balls are replaced by black balls once selected. Initially I have $n$ white balls and $\mu$ samples. I ...