🎲 probability

1. **សេចក្តីថ្លែងបញ្ហា**: គេមានព្រឹត្តិការណ៍ A និង B មិនទាក់ទងគ្នា ដែល $P(A) = x$, $P(B) > y$, និង $0 \leq x < y \leq 1$។ គេដឹងថា $P(A \cup B) = 0.56$ និង $P(A \cap B) = 0.09$។ ត្រ

1. **State the problem:** We have a set of outcomes with probabilities and two events: - Event A: "the outcome is a divisor of 1"

1. **State the problem:** We have outcomes with probabilities and two events: - Event A: outcome is a divisor of 1

1. **State the problem:** We have outcomes 1, 2, and 3 with probabilities 0.2, 0.6, and 0.2 respectively. We define event $A$ as "the outcome is a divisor of 3" and event $B$ as "t

1. **State the problem:** We have outcomes 1, 2, and 3 with probabilities 0.5, 0.4, and 0.1 respectively. We define event A as "the outcome is a divisor of 3" and event B as "the o

1. **State the problem:** We have a probability table with outcomes and their probabilities. We define two events: - Event A: The outcome is a divisor of 3.

1. Let's start by stating the problem: constructing a conditional distribution of a random variable $Y$ given another variable $X=x$. 2. The conditional distribution of $Y$ given $

1. **State the problem:** We have a table showing the distribution of students by gender and major with some missing probabilities. We need to complete the table and find certain p

1. **State the problem:** We roll a 6-sided die and want to find the probability that the outcome is greater than 2. 2. **Formula:** Probability of an event $E$ is given by

1) Problem: Given the piecewise function \( f(x) = \begin{cases} cx^2, & 0 < x < 3 \\ 0, & \text{elsewhere} \end{cases} \), find the constant \( c \) such that \( f(x) \) is a dens

1. **Problem Statement:** We are given that 35% of employees who apply for promotion are successful. Six employees apply, and we want to find the probability that at most four empl

1. **State the problem:** Airreen bought 5 packs of nasi lemak costing 15 in total. She has 3 notes of 10, 5 notes of 5, and 5 notes of 1. She picks two banknotes at random one aft

1. **Problem statement:** Given the density function of a random variable $X$ as $f(x) = kx(2-x)$, find the constant $k$, the $r$th moment, the mean, and the variance. 2. **Step 1:

1. **Problem statement:** We have a standard 52-card deck. A card is drawn, replaced, and then another card is drawn. We want to find the probability of:

1. Problem: Find the probability for two drawings of 2 balls each from a bag with 8 red and 6 blue balls, with replacement, where the first draw is 2 red balls and the second draw

1. **Problem statement:** A bag contains 5 white, 7 black, and 4 red balls, totaling $5 + 7 + 4 = 16$ balls. We draw 3 balls at random. We want to find the probability that all 3 d

1. **Problem Statement:** We are given a Minor Baseball team with 9 starting players, and we want to analyze probabilities related to the order in which players bat.

1. **Problem Statement:** A company has three machines A1, A2, and A3 producing 20%, 35%, and 45% of total output respectively. Defective rates are 2% for A1, 3% for A2, and 5% for

1. **Problem Statement:** We pick two cards simultaneously from a 52-card deck. Define $X$ as the number of face cards (Jack, Queen, King) in the hand and $Y$ as the number of 10s

1. **Problem statement:** Show that $\mathrm{Cov}[aX, Y] = a \mathrm{Cov}[X, Y]$ where $a \in \mathbb{R}$. 2. **Recall the definition of covariance:**

1. **Problem statement:** Given a random variable $X$ with expected value $E[X] = 5$ and variance $\text{Var}(X) = 2$, find $E[X^2]$. 2. **Recall the formula for variance:**