🎲 probability

1. **Problem Statement:** An insurance company receives on average two claims per week from a factory. The number of claims follows a Poisson distribution with parameter $\lambda =

1. **Stating the problem:** We are given a function of the form $$f(x|r) = C(r)(1-x)^{a(r)-1} x^{b(r)-1}$$ where $$r > 0$$ and $$x \in (0,1)$$.

1. **Problem statement:** A speaks the truth 3 out of 4 times, B speaks the truth 7 out of 10 times. They agree that a white ball has been drawn from a bag of 6 differently colored

1. **Problem statement:** A speaks the truth 3 out of 4 times, B speaks the truth 7 out of 10 times. They agree that a white ball has been drawn from a bag of 6 differently colored

1. **Problem:** Given the sample space $\Omega = \{A, B, H\}$ with mutually exclusive events $A, B, H$ and $P(A \cup B) = 0.6$, find $P(H)$. 2. **Formula and rules:** Since $A, B,

1. **State the problem:** Kannika has 9 counters with numbers: 1, 1, 2, 3, 3, 3, 5, 6, 6. She draws two counters without replacement. We need to find the probability that the sum o

1. **State the problem:** We want to find the probability of rolling a 1 or 2 on a fair six-sided die and flipping tails on a fair coin in one trial. 2. **Identify the events:**

1. **State the problem:** We have a spinner with four equally likely spaces: A, B, C, and D. We want to find the probability of spinning a B three times in a row. 2. **Formula and

1. **Problem Statement:** We have a standard deck of 52 cards. We need to write each event as a set and compute its probability.

1. مسئله: یک تاس پرتاب می‌کنیم. اگر عدد روی تاس زوج باشد، یک سکه پرتاب می‌کنیم. اگر عدد فرد باشد، تاس دیگری پرتاب می‌کنیم. 2. فضای نمونه چیست؟ فضای نمونه مجموعه تمام نتایج ممکن یک

1. Masalah: Diberikan masa pakai baterei yang berdistribusi Weibull dengan parameter shape $\alpha=\frac{1}{2}$ dan scale $\beta=2$. Kita diminta mencari: a. Nilai harapan masa pak

1. **State the problem:** We want to find the probability that when a six-sided die is rolled six times, the number 1 appears exactly once. 2. **Formula used:** This is a binomial

1. **Problem Statement:** You provided formulas and properties for eight different probability distributions: Binomial (two versions), Poisson, Uniform discrete, Uniform continuous

1. **State the problem:** We need to find the variance of the power expended by the student's game console per year. 2. **Recall the power function:** The power is given by $$P = 4

1. **Problem statement:** We have 3 X-ray machines, each can be broken or not. Let $A_i$ be the event that machine $i$ is broken. We want to express the following events using $A_1

1. **State the problem:** We have an urn with 10 red balls and 8 white balls. One ball is drawn at random. We need to find the probability that the ball drawn is white. 2. **Formul

1. **State the problem:** We have 50 tickets numbered from 1 to 50. One ticket is drawn at random. We need to find the probability that the drawn ticket has a prime number. 2. **Re

1. **Problem statement:** Two unbiased dice are thrown. Find the probability that the total of the numbers on the dice is greater than 10. 2. **Formula and rules:** The probability

1. **Problem statement:** Two players, A and B, each throw a pair of dice. Player A throws a total of 9. We need to find the probability that player B throws a higher total than 9.

1. **Problem statement:** Given two events A and B with probabilities $P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$, and $P(AB) = \frac{1}{6}$, find the following probabilities: 2. **

1. **Problem statement:** We have two bags, \(B_1\) with 2 white and 1 black ball, and \(B_2\) with 1 white and 2 black balls. One bag is chosen at random, then two balls are drawn