🎲 probability

1. The basic probability formula is given by: $$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

1. The problem asks to find the experimental probability of rolling a 1 on a fair six-sided die after a million rolls, given the theoretical probability is $\frac{1}{6}$.\n\n2. The

1. **State the problem:** Find a stationary distribution $\pi$ for the Markov chain with transition matrix $$

1. Stating the problem: We have 255 students studying history (HIS), geography (GEO), and mathematics (MATH) with given numbers of students studying each and their intersections. W

1. **State the problem:** We have a spinner numbered 1 to 10 with equal probability for each number. We want the probability that the spinner lands on an even number every time in

1. **Problem statement:** Given the joint PDF of random variables $X$ and $Y$ as $$f(x,y) = \frac{2}{75}(2x + y^2), \quad 0 \leq x \leq 3, 1 \leq y \leq 2$$ Find: (1) $P[1 \leq X \

1. **State the problem:** We have a total of 10,000 boxes of bananas from Ecuador and Honduras with the following counts: - Ecuador: 6,000 boxes (200 damaged, 840 overripe)

1. **Problem 02: Two cards drawn with replacement** We want conditional probabilities about the second card given the first card.

1. **Problem 02: Probability with card selections** (a) Find the probability the second card is a face card given the first card was red.

1. Énoncé du problème : Nous avons une entreprise fabriquant des téléviseurs avec 95 % d'appareils en état de fonctionnement (événement $F$). Chaque appareil fait un test d'accepta

1. **Problem 2.3:** Find the probability that, when selecting 3 balls from a box containing $a$ white, $b$ black, and $c$ red balls, at least two balls are of the same color. 2. To

1. **Problem statement:** We have a biased coin that comes up heads with probability $p$. The coin is flipped repeatedly. We want to find the probability that the first tail appear

1. **Problem statement:** We want to find the probability that an insured who did not claim last year will not claim in the next 5 years. 2. **Define states:** Let:

1. **Stating the problem:** We want to find two probabilities related to significant earnings growth and interest rates changes in 2025: (a) The overall probability of significant

1. سوال: احتمال رخ دادن هر یک از پیشامدهای داده شده هنگام پرتاب یک سکه و یک تاس را بیابیم. 2. تعداد کل حالت‌های ممکن را پیدا می‌کنیم:

1. **State the problem:** We have a weather system where the probability of rain tomorrow depends on whether it rained today and yesterday. We are asked to show this can be analyze

1. **Problem Statement:** We are given weather-dependent probabilities for rain based on the last two days' weather, and asked to model this as a Markov chain. We must identify the

1. **Stating the Problem:** We have 3 email accounts with different probabilities of receiving messages and different probabilities of those messages being spam. We want to find th

1. **State the problem:** We have 3 email accounts with different probabilities for receiving messages and spam rates. We want to find the probability that a randomly selected mess

1. **State the problem:** We roll a fair dice twice.

1. **Stating the problem:** We want to find the probability of drawing a club on the first draw and a Jack on the second draw, without replacement, from a standard 52-card deck. 2.