🎲 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. **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