🎲 probability

1. **Problem Statement:** Find the probability density function (pdf), cumulative distribution function (cdf), and expected value of the payment random variable $Y = (X - d)^+$, wh

1. **State the problem:** We have a bag of sweets with two flavours: strawberry and orange. 2. **Given:** The probability of picking a strawberry sweet is $0.8$.

1. Diketahui: - Peluang istri menonton TV sendiri $P(I) = 0.75$

1. Diketahui dua kejadian A dan B dengan probabilitas: $$P(A) = \frac{1}{4}, \quad P(B) = \frac{1}{5}, \quad P(A \cap B) = \frac{1}{20}$$

1. **State the problem:** We want to model a population where about $40\%$ of birds have a wingspan greater than 10 inches using digits 0 to 9. 2. **Understand the modeling:** The

1. **State the problem:** We want to find the probability that a randomly chosen occupied shell is from the Bembicium type. 2. **Identify the data:**

1. **State the problem:** We have a two-way table showing the number of students by grade and their preferred Spirit Day theme (Decades or Pajama). We need to find the probabilitie

1. **State the problem:** We have a two-way table showing the number of students who used search engines A and B and whether they passed or failed an assignment. We want to find va

1. **State the problem:** We want to find the expected value of the game given the probabilities and payoffs. 2. **Recall the formula for expected value:**

1. **State the problem:** We want to find the expected value of the game where you draw one marble from a bag containing 3 gold, 10 silver, and 26 black marbles. 2. **Total number

1. **State the problem:** We have a bag with $3$ red balls, $4$ white balls, and $2$ blue balls. We want to find the probability of randomly selecting a red ball. 2. **Formula for

1. **State the problem:** We have a bag with $3$ red balls, $4$ white balls, and $2$ blue balls. We want to find the probability of randomly selecting a red ball. 2. **Formula for

1. **Problem:** Find the probability that both balls drawn without replacement from a box containing 8 red, 5 blue, and 3 green balls are red. 2. **Step 1: Understand the problem.*

1. **Problem:** Find the probability that both balls drawn without replacement from a box containing 8 red, 5 blue, and 3 green balls are red. 2. **Formula:** Probability of both e

1. **Problem Statement:** Two players, A and B, each start with $2$ dollars. They play a game betting $1$ dollar each round until one player is broke (has $0$ dollars). Player A wi

1. **State the problem:** We are given probabilities for two events: getting an award for design ($P(D) = 0.16$), getting an award for efficient use of materials ($P(M) = 0.24$), a

1. ปัญหาคือการหาจำนวนผลลัพธ์ทั้งหมดที่เป็นไปได้เมื่อโยนเหรียญ 1 เหรียญ 3 ครั้ง 2. สูตรที่ใช้คือจำนวนผลลัพธ์ทั้งหมด = $2^n$ โดยที่ $n$ คือจำนวนครั้งที่โยนเหรียญ

1. The problem is to verify the expression for the marginal density function $f_X(x)$ given by $$f_X(x) = \int_\mathbb{R} f(x,y) \, dy = \int_x^\infty cx(y-x)e^{-y} \, dy.$$

1. **Problem statement:** Two dice are rolled. Define events: - $A$: sum of two dice equals $3$

1. **State the problem:** We have two independent and identically distributed random variables $X$ and $Y$ with pdf $f(x) = e^{2-x}$ for $x > 2$ and 0 otherwise. We want to find th

1. **Problem Statement:** We have two phases of repair for each machine: