Articles of probability

Should you ever stop rolling, THE SEQUEL

Inspired by this question, I want to know if there is a version of the scenario that actually fits Newb’s intuition about the problem. Scenario template You roll a 6-sided die and add up the cumulative sum of your rolls. The game ends under the following conditions, with the associated payouts: You choose to stop […]

The probability two balls have the same number

Suppose I have $10^6$ jars, and $k$ balls are randomly and independently placed in each jar. I am given that the probability that there exists a jar with 2 balls is approximately $50\%$. Then $k$ is: $500000$ $64$ $1+2+\ldots+64$ $\approx 64\log(64)$ So my way of thinking is this. $k$ balls are given a random number […]

Probability of two events

I am having trouble of calculating the following probability: Let $\epsilon_i$, $i=1,\dotsc,N$ be Rademacher random variables. Let $n_i\in \{0, 1, 2, \dotsc, M\}$, $i=1,\dotsc,N$ such that $\sum_{i=1}^Nn_i=M$. I want to calculate $$ P\left(\left\{\prod_{i=1}^N\epsilon^{n_i}_i=1\right\}\bigcap\left\{\sum_{i=1}^N\epsilon_i=0\right\}\right). $$ Thank you.

Variance of the sums of all combinations of a set of numbers

Let’s assume a finite set of $n$ real numbers: $$\mathbb{V}=\{a,b,c,…,z\}$$ Now we take all the possible combinations of this set, including $0$: $$\mathbb{C}=\{\{0\},\{a\},\{b\},…\{a,b\},\{a,c\},…,\{a,b,c,…,z\}\}$$ After that we sum up all of those combinations: $$\mathbb{S}=\{\{0\},\{a\},\{b\},…\{a+b\},\{a+c\},…,\{a+b+c+…+z\}\}$$ We now have $2^n$ sums in $\mathbb{S}$. My question What is the variance of $\mathbb{S}$? What have I found out and tried […]

Cumulative Distribution Function of Sum two Weibull random variables

If $Y_0$ and $Y_1$ both have Weibull distribution i.e. $Y_0 \sim Weibull(\lambda_0,\beta_0)$ and $Y_1 \sim Weibull(\lambda_1,\beta_1)$ then what will be cumulative density function of $Y_0+Y_1$, i.e. $$\Pr(Y_0+Y_1<y)=\int_0^\infty \Pr(Y_0<y-z|z=Y_1)f_{Y_1}(Y_1)\ dY_1$$

clarification on Chernoff's inequality

i’m studing probabilistic algorithms: the ones that – with a great gain in efficency – sometimes could return a false response. They return the true response with a probability of $\frac{3}{4}$. The the way to use them to lower the error probability is: runs $t$ times, compare results and gets the more frequent answer. Those […]

Creating unusual probabilities with a single dice, using the minimal number of expected rolls

Problem I want to create an ‘event’ with probability of $\frac{1}{7}$ with a single dice as efficiently as possible (to roll the dice as little as possible). To give you some better understanding of the question, if I would like an event with probability of $\frac{1}{9}$, I could easily do it in various ways. One […]

Expected number of dice rolls of an unfair dice to roll every side equally many sides

I am having trouble with solving the following problem: The probability that a $d$-sided dice lands on its $k$th side is equal to $p_k$ for $k\in \{k\in\mathbb{N},k≤d\}$ and $p_1+p_2+p_3+…+p_d=1$. Roll this dice (at least once) until every side is rolled equally many times. Find a function $F(p_1,p_2…)$ which gives the expected number of rolls $n$ […]

Uniform distribution on unit disk

Let $(X, Y)$ be a random point chosen according to the uniform distribution in the disk of radius 1 centered at the origin. Compute the densities of $X$ and of $Y$. I know that the joint density of $X$ and $Y$ is $\frac{1}{\pi}$ since when we integrate $\frac{1}{\pi}$ over the unit circle, we get $1$. […]

Generating function: Probability regarding coin toss

If a coin is flipped 25 times with eight tails occurring, what is the probability that no run of six (or more) consecutive heads occur? Wasn’t sure how to approach this and am quite positive my generating function is incorrect. My attempted work: Consider $e_H,e_T$ s.t $e_H$ denotes the number of times our coin lands […]