Sent to me by a friend, I’m stumped. We have an urn. It is first filled with balls of $C$ different colors, ${N_1,N_2,…N_c}$ of each of the colors. Lastly, a ball with a distinct marking is added. We are not given $C$ or any of the $N$ values. A machine removes the balls without replacement […]

Suppose we throw $k$ balls into $n$ bins. Assume that $\log^2n\le k\le n$. Is there a high probability bound (preferably exponential) on the number of occupied (i.e., non-empty) bins? Something like: $\Pr(\text{Num of Occupied Bins} < \alpha k) < e^{-O(k)}$. Thanks!

Assume that there are $m$ balls and $n$ urns with $m\gt n$. Each ball is thrown randomly and uniformly into urns. That is, each ball goes into each urn with probability $\dfrac1n$. What is the probability that there are exactly $r$ urns with at least one ball in it? In other words, what is the […]

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 urns $1, 2, . . . , i$. Find the expected number of urns that are empty. Can somebody help […]

Well, task is in the title. I need number of ways to place $k$ balls in $F$ boxes where exactly $r$ boxes contain exactly one ball. It means that exactly $r$ boxes should contain exactly one ball and another $F – r$ boxes should contain some number from set $ S = \{\mathbb{N}\setminus\{1\}\} \cup\ \{0\} […]

I asked a somewhat related question recently and then became interested in this one: how many people are required, on average, until 3 share a birthday? More generally, if we have $M$ bins, what is the expected number of balls we must toss before some bin contains exactly 3 balls? The straightforward techniques used in […]

Given a set $S$ such that $|S|=n$, A random item is chosen randomly from $S$, and being appended to a new set $T$. This process is being repeated $n$ times (with repetition), what is the expected value of $|T|$ ?

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 element may be picked more than once, but counted as only one “marked element”. The PRNG is properly seeded. […]

I have $n_1$ red balls in a box $A$. These balls are numbered from $1, \cdots n_1$. Let make a copy version of box $A$, called box $D$ (It means that the box $D$ will contain $n_1$ red balls from $1, \cdots n_1$). Throw these balls to a box $B$ with loss probability $p$. Now, […]

Assume that there are $n$ people line up to get into $m$ different clubs. How many ways to do it if each people is different and order of people in the line matters. Sol. Clearly, each people has $m$ different choices for clubs. So there are $m^n$ ways of choosing clubs. Now we count number […]

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