🎲 probability

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:**

1. **Problem statement:** We have a random variable $X$ with values $-1, 0, 1$ and probabilities $\frac{1}{8}, \frac{2}{8}, \frac{5}{8}$ respectively.

1. **Problem statement:** We have a random variable $X$ with values between 0 and 1 and probability density function (pdf) $f(x) = 2x$. We want to compute the variance of $X$, deno

1. **State the problem:** We want to compute the expected value $E[X]$ and variance $\mathrm{Var}(X)$ of a discrete random variable $X$ with the given probability mass function (pm

1. **Problem Statement:** We roll two special 4-sided dice, each showing values from 1 to 4. Define $X$ as the minimum of the two values and $Y$ as the maximum of the two values. (

1. **Problem statement:** Chris tries to throw a ball into a basket with success probability $p=\frac{1}{3}$ each attempt, independent of others.

1. **Problem Statement:** A student rolls a die repeatedly until the first "4" appears. Define $X$ as the number of rolls needed to get this first "4" (including the roll that show