🎲 probability

1. **State the problem:** We are given a discrete random variable with scores and their corresponding probabilities. We need to find the expected value (mean) and the standard devi

1. **Problem Statement:** Given the table of grades by gender, we need to find various probabilities related to the students. 2. **Total number of students:** $$74$$ (given in the

1. **Problem Statement:** We want to find the probability $P(X + Y \leq 5)$ for a continuous bivariate random variable $(X,Y)$ with joint density $g(x,y)$ defined over $0 < X < 4$

1. **Problem statement:** We want to find the correct integral expressions for the probability $P(X \leq 2 \cup Y \leq 2)$ where $(X,Y)$ is a continuous bivariate random variable w

1. **Problem Statement:** You start with 3 tokens and want to reach 5 tokens before hitting 0. Each turn, you bet the maximum tokens possible but not more than needed to reach 5 if

1. **State the problem:** You roll three fair 6-sided dice. The payouts are: - $20 if all three dice show the same number.

1. **Problem statement:** We have two factories, A and B, producing red and black widgets with given probabilities. We select one factory uniformly at random, then sample two widge

1. **Problem Statement:** We want to understand the Poisson distribution, which models the number of events occurring in a fixed interval of time or space when these events happen

1. The problem is to find the moment generating function (MGF) of the binomial probability distribution. 2. The binomial distribution models the number of successes in $n$ independ

1. **Problem Statement:** We are given that 30% of students offered English, 20% offered Hindi, and 10% offered both English and Hindi. We need to find the probability that a rando

1. **Problem Statement:** A number is selected at random from the first thirty natural numbers (1 to 30). We need to find the probability that the number is a multiple of either 3

1. **Problem statement:** A card is drawn at random from a standard deck of 52 cards. We want to find the probability that the card is neither a spade nor a Jack. 2. **Formula for

1. **Problem statement:** A bag contains 4 red balls, 6 blue balls, and 8 pink balls. One ball is drawn at random and replaced with 3 pink balls. We want to find the probability th

1. **Problem Statement:** We have tickets numbered from 1 to 20. We want to find the probability that a randomly drawn ticket has a number that is a multiple of 3 or 5. 2. **Formul

1. **Problem Statement:** We have a survey about residents reading three magazines: Newsweek (N), Vogue (V), and Elle (E). Given percentages and counts, we need to find: - 117: Num

1. **បញ្ហា**: គេមានកាក់ ៩ ដោយចុះលេខពី ១ ដល់ ៩។ គេចាប់យកកាក់ ៤ ដោយចៃដន្យ ហើយតម្រៀបជាជួរដេកបង្កើតបានលេខ ៤ ខ្ទង់។ រកប្រូបាបដែលចំនួននោះជាចំនួនសេស (គឺចំនួនដែលចែកបានដោយ ២)។ 2. **វិធីដោះស

1. **សេចក្តីថ្លែងបញ្ហា**: នៅលើក្រដាសកាតុង ១០០ សន្លឹក មានលេខរៀងពី 1 ដល់ 100។ គេចាប់យកក្រដាសមួយសន្លឹកដោយចៃដន្យ។ រកប្រូបាបដែលលេខចុះលើក្រដាសនោះចែកដាច់នឹង 5។ 2. **រូបមន្តប្រើប្រាស់**: ប

1. **សេចក្តីថ្លែងបញ្ហា**: គេបោះគ្រាប់ឡុកឡាក់ពីរគ្រាប់ព្រមគ្នា។ ត្រូវរកប្រូបាបដែលផលបូកលេខចេញលើគ្រាប់ឡុកឡាក់ទាំងពីរស្មើ ៥ ឬស្មើ ៨។ 2. **កំណត់លក្ខខណ្ឌ និងរូបមន្ត**:

1. **Stating the problem:** We are given two events A and B with probabilities P(A) and P(B), and the formula for the probability of both events occurring together: $$P(A \text{ an

1. **Stating the problem:** We are given probabilities related to two events $A$ and $B$ and asked to determine if they are independent. 2. **Recall the definition of independence:

1. **សេចក្តីថ្លែងបញ្ហា**: គេឱ្យព្រឹត្តិការណ៍ A និង B ដែលមានប្រូបាប P(A) = 0.55 និង P(B) = 0.25។ ព្រឹត្តិការណ៍ទាំងពីរនេះមិនចុះសម្រុងគ្នា (mutually exclusive)។ ត្រូវរកប្រូបាប P(A \cu