🎲 probability

1. **Problem statement:** We have 5 identical cards numbered 2, 3, 4, 5, 6. Two cards are drawn one after the other with replacement, forming a two-digit number. We want to find th

1. **Problem statement:** We have 8 cards numbered 1 to 8. Two cards are drawn successively with replacement. We want to find the probability that the absolute difference between t

1. **Problem Statement:** We have a continuous random variable $x$ with probability density function (pdf):

1. **Problem Statement:** Determine whether the given distribution is a valid probability distribution for a discrete random variable $X$ with values $5, 6, 7, 4$ and probabilities

1. **Problem Statement:** Given the probability density function (pdf) $f(x) = 3x^2$ for $0 \leq x \leq 1$, find values of $a$ and $b$ such that: i) $P(X \leq a) = P(X > b) = 0.05$

1. **State the problem:** We want to find the probability that a randomly chosen customer who ordered a dessert did not order a starter. 2. **Identify the sets and numbers:**

1. **State the problem:** We want to find the probability that a randomly chosen customer who ordered a dessert did not order a starter. 2. **Identify the sets and numbers:**

1. **Énoncé du problème :** On considère une variable aléatoire $X$ suivant une loi du Khi-deux avec $k=4$ degrés de liberté.

1. ปัญหาคือ ต้องการหาความน่าจะเป็นที่จะสุ่มนิสิต 5 คน จากนิสิตทั้งหมด 16 คน โดยที่ได้นิสิตชายอย่างมาก 2 คน 2. สูตรความน่าจะเป็นสำหรับการเลือกแบบไม่เรียงลำดับคือ

1. ปัญหาคือการหาค่าคาดหวัง (Expected Value) และความแปรปรวน (Variance) ของตัวแปรสุ่ม X 2. สูตรค่าคาดหวังคือ $$E(X) = \sum xP(x)$$ ซึ่งเป็นผลรวมของค่าตัวแปรคูณกับความน่าจะเป็นของแต่ล

1. The problem is to find the expected value $E(X)$ and the variance $\mathrm{Var}(X)$ of a random variable $X$ given its probability distribution. 2. The expected value formula is

1. **Problem statement:** We have two real numbers $x, y$ chosen randomly from the interval $[0,1]$.

1. **State the problem:** We have 6 batteries in total, 2 of which are defective. We select 2 batteries at random without replacement. We want to find the probability that both sel

1. **Stating the problem:** We want to find the probability that a participant wins the game.

1. **State the problem:** We are given probabilities $P(A_1)=0.58$, $P(A_2)=0.34$, and $P(A_1 \cap A_2)=0.09$. We want to find the probability that at least one of the two diagnose

1. **Problem statement:** We have a binomial random variable $X \sim \text{Binomial}(n=5, p=0.55)$ and want to find the probability $P(X \leq 2)$, which means the probability that

1. **State the problem:** Given probabilities $P(A)=0.792$, $P(B)=0.538$, $P(C)=0.783$, and intersections $P(A \cap B)=0.394$, $P(A \cap C)=0.589$, we want to understand these valu

1. مسئله: مجموعه $A = \{-1, 3, 5, 7, 9, 11\}$ داده شده است. باید احتمال انتخاب تصادفی زیرمجموعه‌ای از $A$ را پیدا کنیم که حاصل ضرب بزرگترین و کوچکترین عضو آن برابر با 7 باشد. 2. اب

1. مسئله: مجموعه $A = \{-13, 5, 7, 9, 11\}$ داده شده است. می‌خواهیم احتمال انتخاب تصادفی زیرمجموعه‌ای از $A$ را که حاصل جمع بزرگترین و کوچکترین عضو آن برابر 7 باشد، پیدا کنیم. 2. ا

1. **State the problem:** We have a spinner with three colored sections: blue, yellow, and pink. It was spun 100 times. 2. **Given data:**

1. **Problem Statement:** We have two boxes: Box 1 contains numbers 3, 5, 8 and Box 2 contains numbers 4, 7. One slip is taken from each box.