🎲 probability

1. **Problem Statement:** Show that the function $P(X) = \frac{X}{10}$ for $X=1,2,3,4$ is a probability mass function (pmf) of a discrete random variable $X$. Then find: i. $P(X \l

1. **State the problem:** Given the cumulative distribution function (CDF) $$F(x) = \frac{1}{1 + e^{-x}}$$ for $$-\infty < x < \infty$$, find the probability density function (PDF)

1. **State the problem:** We want to find the probability of hitting the target at least once when 10 shots are fired independently, and the probability of hitting the target with

1. **Problem statement:** Given a binomial random variable $X$ with expected value $E(X) = 6$ and variance $Var(X) = 2.4$, find the probability $P(X=5)$. 2. **Recall the formulas f

1. ปัญหาคือหาความน่าจะเป็นที่ครอบครัวมีบุตรชาย 2 คน จากทั้งหมด 3 คน โดยที่ทราบว่ามีบุตรชายอยู่แล้ว 1 คน 2. กำหนดให้ \(B\) คือเหตุการณ์ที่มีบุตรชาย 2 คนใน 3 คน และ \(A\) คือเหตุการณ

1. **State the problem:** We have an urn with 4 blue balls and 5 red balls. We want to find the probability of randomly selecting a blue ball. 2. **Formula for probability:** The p

1. **Problem statement:** Given the joint density function $$f(x,y) = k(x^2 + y^2)$$ for $$30 \leq x < 50$$ and $$30 \leq y < 50$$, and zero elsewhere, we need to find several quan

1. **Problem statement:** The survival time after an operation follows an exponential distribution with an average (mean) of 4 years. Given that a patient has already survived 2 ye

1. **Problem Statement:** Find the PDF, expected value, and variance of an exponential random variable $X \sim \text{Exponential}(\lambda)$. 2. **Formula for Exponential Distributi

1. **Problem Statement:** Find the PDF, expected value, and variance of a uniform random variable $U \sim \text{Uniform}(a,b)$. 2. **Formula for Uniform Distribution:**

1. **Problem statement:** A box contains 6 white, 7 red, and 9 black balls. What is the probability of drawing a ball that is either red or white? 2. **Formula and rules:** Probabi

1. **Problem Statement:** Construct the probability distribution of the random variable $G$ representing the number of Germans selected when 3 consuls are chosen at random from 4 A

1. **Problem statement:** We have a point moving on a number line starting at 0. When a die is rolled: - If the result is 1 or 2, the point moves 1 step to the left (negative direc

1. **Problem statement:** Given independent random variables $X_1 \sim N(60,25)$, $X_2 \sim \text{Gam}(3.5,\sqrt{5})$, $X_3 \sim \text{Gam}(6,\sqrt{3})$, and $X_4 \sim N(50,16)$, i

1. The problem asks to solve a probability problem using one of the distributions: hypergeometric, binomial, or Poisson. 2. First, identify the type of problem:

1. **Problem statement:** A library has 20 fiction books and 30 non-fiction books, total 50 books. A student borrows 6 books randomly. We want the probability that less than 6 of t

1. **Problem Statement:** Identify which of the given events are simple events. 2. **Definition:** A simple event is an event that consists of exactly one outcome.

1. **Énoncé du problème :** Un individu essaie un mot de passe au hasard. La probabilité d'être refusé est $\frac{999}{1000}$. L'ordinateur accepte 3 essais avant de couper la conn

1. **Problem Statement:** Find the probability that a randomly selected student is female given the data:

1. **State the problem:** We have a random variable $X$ with values $0, 1, 2, 3$ and corresponding probabilities $0.1, 0.4, 0.3, 0.2$. We need to find the mean (expected value) $\m

1. **Problem Statement:** We flip four coins and define the random variable $X$ as the number of heads in the four tosses. We need to calculate the variance and standard deviation