🎲 probability

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

1. **Problem Statement:** We have a random variable $X$ with probability density function (pdf) $$f(x) = \begin{cases} \frac{1}{6}x, & 2 < x \leq 4 \\ 0, & \text{otherwise} \end{ca

1. **Problem statement:** We have three people: Alice, Bob, and Charlotte, each independently finding a butterfly with probabilities 0.17, 0.25, and 0.45 respectively. Define rando

1. **Problem statement:** We have three independent people (Alice, Bob, Charlotte) each catching a butterfly with given probabilities. Define $X$ as the total number of butterflies

1. **Problem Statement:** We want to find the probability that the vowels in the word \textbf{MATHEMATICS} appear in alphabetical order (A, A, E, I) when the letters are arranged r

1. **State the problem:** Amit, Tan, and Raju shared 36 rambutans. Amit took 12 rambutans, and Tan and Raju took the rest. The number of rambutans taken by Tan is twice the number

1. **Problem Statement:** You are given probabilities related to customer behavior in a marketing campaign:

1. **Problem Statement:** We have a table showing the number of employees classified by age groups and departments. We need to calculate various probabilities based on this data.

1. **Problem Statement:** A manufacturer has 100 memory chips with 4% defective. A sample of 20 chips is selected. Let $X$ be the number of faulty chips in the sample.

1. **State the problem:** We have two boxes with items, some faulty and some not. We want to find probabilities related to selecting faulty or non-faulty items.

1. You mentioned you have two exercises: one on conditional probability and one on random variables. 2. Let's start with conditional probability. The problem typically asks: Given

1. **State the problem:** Shifaa is late 2 days out of 6 working days in a week. We want to find the expected number of days she is late in a week. 2. **Formula used:** The expecte

1. **Problem 1:** A bucket contains 3 red, 4 yellow, and 5 purple balls. One ball is taken without replacement. Find the probability that the first ball is red and the second is pu

1. **Problem 1: Probability of exceeding 13,000 steps** The number of steps per day is normally distributed with mean $\mu=10000$ and standard deviation $\sigma=1500$.

1. **State the problem:** We are given probabilities:

1. The problem is to understand and apply conditional probability formulas. 2. Conditional probability measures the probability of an event $A$ occurring given that another event $