Articles of probability

“The three shooters algorithm”

As requested at I have moved that topic into here: Like the title says, you’ve got 3 ‘shooters’: shooter A, B and C. The rules are the following: Each one has a different chance to hit: Shooter A will hit for 100% of the times. Shooter B will hit for 80% of the times. […]

How to find minimum and which combinations to guarantee minimun hits on lotto given N picked numbers?

I have been reasearching the internet for a few days now trying to figure out what we call here in México “Ruedas o Reducciones” of combinations, these are simple ways of reducing the numbers of tickets you need to buy to get “the most” out of your money. Basic example: In the lotto I play […]

Two different normal distributions – probability A > B

If I take a single sample A from a normal distribution with mean M1 and variance V1 then take a sample B from a normal distribution with mean M2 and variance V2, then what is the probability that A>B ? I dare say this question has been asked and answered many times, but I don’t […]

how to graph Z on the (x, y) plane and how to find the cdf of Z?

We consider a random point $(X, Y)$ chosen with the following joiint density $$f(x,y) = \begin{cases} \frac{1}{x^2y^2}, & \text{if } x \ge 1 \text{ and } y \ge 1 \\[2ex] 0, & \text{else} \end{cases}$$ Consider another random variable Z=XY. A. for $z \gt 1$ make a plot of the region in the $(x, y)$ plane […]

Tossing n balls into n bins and proving independence

Suppose we toss $n$ balls into $n$ bins with all outcomes equally likely. For clarity, assume the balls each have a unique number written on them, from $1$ to $n$, and also assume the bins are numbered from $1$ to $n$. Let $D$ be the event that all of the n balls land in different […]

Limiting distribution of the sum of squared independent normals

Let $t\in[0,1]$ and suppose $Z_t$ is a standard normal random variable, such that $Z_t$ and $Z_s$ are independent if $s\neq t$. If we pick $0\leq t_1\leq \ldots\leq t_k\leq 1$ then $$X = \sum_{j=1}^k Z_{t_j}^2$$ is a chi-squared distribution with $k-1$ degrees of freedom. What happens in the limit? What is the distribution of $$Y = […]

Find prob. that only select red balls from $n$ (red+blue) balls

There are 4 blue balls and 6 red balls(total 10 balls). $X$ is a random variable of the number of selected balls(without replacement), in which $$P(X=1)=0.1$$ $$P(X=2)=0.5$$ $$P(X=3)=0.2$$ $$P(X=4)=0.1$$ $$P(X=10)=0.1$$ Then, what is probability of only selecting red balls? This is what I have tried: The (conditional) probability that all $r$ of the balls are […]

Conditional probability on zero probability events (Application)

This question can be regarded as an application to this question. Let $(\Omega ,{\mathcal {F}},P)$ be a probability space, $Z_1$ is a $(M_1,{\mathcal {M}_1})$-value random variable (that is it is $(\mathcal{F},\mathcal{M}_1)$ measurable), $Z_2$ is a $(M_2,{\mathcal {M}}_2)$-value random variable, $W$ is a $(N,{\mathcal {N}})$-value random variable, and $g$ is a measurable function. Assume that $Z_1$ […]

Probability: 5 cards drawn at random from a well-shuffled pack of 52 cards

A poker hand consists of 5 cards drawn at random from a well-shuffled pack of 52 cards. Then, the probability that a poker hand consists of a pair and a triple of equal face values (for example, 2 sevens and 3 kings or 2 aces and 3 queens, etc.) is? I’m a bit confused how […]

Flip coin Problem Combinatorics

Let’s say we are flipping a coin $n$ times. What is the probability that we always have more heads than tails. Meaning that if we are counting the number of times we have had heads and the number of time we have had tails, what is the probability that throughout the $n$ throws we continuously […]