🎲 probability

1. **Problem Statement:** We have a continuous random variable $X$ representing the total number of hours (in units of 100 hours) a family runs a vacuum cleaner in a year. The prob

1. **State the problem:** A bag contains 8 identical red balls and an unknown number of white balls. The probability of picking a red ball initially is given as $\frac{5}{2}$, whic

1. **Stating the problem:** A bag contains 8 identical red balls and an unknown number of white balls, say $w$. The probability of drawing a red ball initially is given as $\frac{1

1. **Problem statement:** We have a random variable $X$ representing the shelf life of medicine bottles with density function $$f(x) = \frac{20000}{(x+100)^3} \text{ for } x>0, \te

1. **Problem (a):** Find the cumulative distribution function (CDF) values for $F(x) = \frac{1}{16} \left( \frac{4}{x} \right)$ at $x=0,1,2,3,4$. 2. **Important note:** The functio

1. **State the problem:** Penelope rolls four regular six-sided dice (each numbered 1 to 6) and sums the dots on the top faces. We want to find how many different possible sums she

1. **Problem:** A box contains 10 red, 20 blue, and 30 green marbles. 5 marbles are drawn. Find: (i) Probability all 5 are blue.

1. **Problem:** Cho A, B là hai biến cố xung khắc. Tìm đẳng thức đúng liên quan đến xác suất của A và B. 2. **Formula:** Với hai biến cố xung khắc, ta có công thức:

1. **Problem Statement:** Draw a tree diagram to represent the experiment where two cards are drawn without replacement from nine cards numbered 1 to 9, noting whether each card is

1. The problem asks to draw a tree diagram representing the experiment of drawing two cards without replacement from a box containing cards numbered 1 to 9, noting whether each car

1. **Problem:** A jar contains 12 red balls, 8 blue balls, and 10 green balls. Two balls are picked without replacement. Find the probability both balls are the same color. 2. **Fo

1. The problem is to find the expected value $E(X)$ given by the integral: $$E(X) = \int_{0}^{10} x \cdot \left[ \frac{1}{80}(x + 3) \right] \, dx$$

1. **Problem Statement:** We have 500 students surveyed about their preferences for watching football, hockey, and basketball. Given percentages for each sport and their pairwise i

1. **Problem statement:** Bạn Nam chọn 6 số từ tập $S=\{21,22,23,24,25,26,27,28,29\}$ và xếp vào vị trí $A,B,C,M,N,P$ sao cho các bộ ba số ở vị trí $(A,M,B)$, $(B,N,C)$, $(C,P,A)$

1. **Problem Statement:** From a lot of 30 bulbs including 6 defective bulbs, a sample of 2 bulbs is drawn at random one by one with replacement. Find the probability distribution

1. **Problem statement:** Alex rolls an 8-sided die repeatedly and stops once he rolls two consecutive 8s. We want to find the expected number of rolls until he stops. 2. **Define

1. مسئله: یک زنجیره مارکف با فضای حالت $S=\{0,1,2,3,4\}$ داریم که احتمال انتقال از حالت $r$ به $r+1$ برابر $\frac{2}{3}$ و به $r-1$ برابر $\frac{1}{3}$ است برای $r=1,2,3$. حالت‌های

1. The problem asks to define the sample space for a given experiment. 2. The sample space is the set of all possible outcomes of an experiment.

1. The problem is to understand the probability of selecting a specific letter from the set {C, S, M, B, Q, T}. 2. Probability is calculated as the number of favorable outcomes div

1. **Problem statement:** We are given that on average, 1 out of 10 telephones is busy, so the probability a telephone is busy is $p=\frac{1}{10}=0.1$. We select 6 telephones at ra

1. Masalah ini menanyakan jumlah minimum kelereng yang harus diambil dari kantong yang berisi 10 kelereng merah, 10 kelereng biru, dan 10 kelereng putih agar dijamin memiliki setid