Articles of probability

How to find expected solved variables from a linear equation

I have a difficult problem about probability that needs your help. Let $(x_1,x_2,x_3,x_4)$ be four unknown variables. From these variables, we can created a set of equations as follows: $$ \begin{pmatrix} x_1 & & & \\ & x_2 & & \\ & & x_3 & \\ & & & x_4\\ x_1 +x_2& & & \\ […]

Geometrical Probability

$O(2,3)$, $A(2,0)$, $B\left(1,\dfrac{1}{\sqrt{3}}\right)$ are the vertices of $\Delta{OAB}$ on the $\text{x-y}$ plane. Let $\text{R}$ be the region consisting of all points $P$ inside the triangle, which satisfy: $$d(P,OA)\geq \min\{d(P,OB), d(P,AB)\} $$ For a random distribution of point P, the probability that it lies in the region $\text{R}$ is of the form: $a-b\sqrt[c]{d}$ $\text{Find:}d^{a}+ c^{b}$$$$$ $d(X,YZ)$ […]

How to explain why the probability of a continuous random variable at a specific value is 0?

Consider X as a continuous random variable which can assume any value in [0, 1]. It is known that P(X=x)=0 where P is the probability density function. I want to understand this intuitively. The math insight article helps me somewhat: In other words, the probability that the random number X is any particular number x∈[0,1] […]

Classic 10 Balls into 4 boxes problem

Question:Throw at random 10 balls into 4 boxes. What is the probability that exactly 2 boxes remain empty? My Solution: Using the stars and bars method, the boxes and balls can be thought as $0$ and | e.g. one configuration would be $$ 000000|00|0|0 $$ which would represent 6 balls in the first box etc. […]

Probability that team $A$ has more points than team $B$

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The […]

Roll 6 dice, find the number of outcomes with 3 distinct numbers.

Suppose you roll six dice, how many outcomes are there with 3 distinct numbers. My attempt: First there are ${6 \choose 3}$ ways to choose these 3 distinct numbers. We consider 3 cases; Case 1: 3 pars of repeated numbers e.g. $223344$. There are ${3\choose 3}$ choices for the values, and for the ordering there […]

Simple probability problems

This may be very simple, but I need to know correct answers. Problem 1. There are $6$ balls in the box – $2$ black, and $4$ white. We take $3$ balls from the box. What is the probability that we took exactly 1 black ball. Problem 2. What is the probability to score $7$ points […]

Determine the limit distribution

I have this question here that I could use some help with. Let $X_1$, $X_2$, . . . be a sequence of random variables such that $P(X_n=\frac{k}{n})=\frac{1}{n}$, for $k=1,2,…,n$ Determine the limit distribution of $X_n$ as $n\rightarrow \infty$. Now I think that if I could find $f_X(x)$ and $F_X(x)$ for $X_n$ then I would have […]

Why is Cumulative “Density” wrong?

CDF stands for cumulative distribution function. However, it is “loosely” referred to as Cumulative Density many times. As i write this question, I have a suggestion toolbar on this page that lists over 10 questions with the words “Cumulative Density” in them. I came across this question in this forum post where a comment clearly […]

Probability of picking marbles from a bag with only the ratio of marbles given

Here is a question that is puzzling me: A bag contains a large number of marbles; the numbers of the red, blue and yellow marbles are in the ratio $3:4:5$. Four marbles are randomly drawn without replacement. What is the probability that you will draw $1$ red marble, $2$ blue marbles and $1$ yellow marble? […]