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Overall problem: Sample i.u.d. from $\{1,\dots,n\}$. What is (a good lower bound for) the probability of getting the values $1$ and $2$ before either you get a number you have seen before or you have …

Consider two positive integers $x \ne y$ and let $n = max\{\lfloor \log_2{x} \rfloor +1 ,\lfloor \log_2{y} \rfloor +1 \}$. Choose a prime $p$ randomly from the first $3n$ primes. What is the prob …

Say we toss $d$ pairwise independent coins, each with probability $1/d$ of getting a head. What is the highest lower bound one can give for the probability of getting exactly one head? If they had bee …

If you sample $n$ integers from the range $1$ to $n$ inclusive it seems intuitive that you are likely to get a lot of numbers exactly once. Call $X_n$ the number of integers you get that occur exactl …

If you toss a coin $2\ell-1$ times you get a sequence of outcomes, say, $HTHTHTH$ for $\ell = 4$. I am trying to work out the probability that there are at least two runs (in other words contiguous s …

If you sample $n$ vectors each with $m$ entries, with each entry chosen from the set $\{-1, 1\}$, how can you calculate the expected maximum absolute value of the inner product between all pairs of ve …

I am not a professional mathematician so please excuse me if my question is not phrased correctly. I am interested in the following simple sounding problem. Consider a random $n$ by $n$ $0$-$1$ matr …

Let $S_1,\dots, S_n$ be Bernoulli random variables which are $4$-wise independent. We have that for each $i$, $P(S_i = 1) = p$ for some fixed probability $0 < p < 1$. What can we say about $P(\fora …

For large and even $n$ consider a random degree $n$ polynomial $v(t)$ with coefficients from $\{-1,0,1\}$. The coefficients are chosen uniformly and independently. Is it possible to get an estima …

Consider $d$ dimensional space cut by $n$ hyperplanes in general position, each one of which goes through the origin. The number of distinct regions created is known to be: $$2\sum_{i=0}^{d-1} {n -1 …

Imagine you sample $n$ number with replacement uniformly from the integers $1,\dots, n$. Let $X$ be the minimum of these samples. I am interested in $\mathbb{E}(X)$ but with a twist. All I know is t …

Throw $n$ balls into $n$ bins, and let $X_n$ be the max load. That is the number of balls in the fullest bin. It is known that if the balls are thrown uniformly and independently at random then $\mat …

Imagine you sample $n$ numbers with replacement uniformly from the integers $1,\dots, n$ (we can assume $n$ is large). Let $X$ be the minimum of these samples. I am interested in $\mathbb{E}(X)$ but …

Say we toss $d$ $k$-wise independent coins, each with probability $1/d$ of getting a head. What is the highest lower bound one can give for the probability of getting exactly one head? In the first i …

If $X_i$ is a sequence of $d$ dimensional i.i.d. integer valued random vectors with covariance matrix $\Sigma$ and $\mathbb{E}(X_i) = \mu$. Let each element of $X_i$ be chosen i.u.d. from $\{-1,1\}$. …