🎲 probability

1. **Problem statement:** Customers arrive at a bank at a rate of 10 per hour. Each customer is new or existing with probability 0.5. Given exactly 10 new customers arrived in one

1. Problem 4a: A bag contains 2 white, 3 black, and 4 red balls. Find the probability of drawing: i. White or red ball

1. **Problem statement:** We have two boxes U1 and U2 with colored balls. U1 contains 2 red and 3 green balls, U2 contains 3 red and 2 green balls. We draw 3 balls as follows: - Dr

1. **بيان المسألة:** نقوم بسحب 3 كرات من صناديق U1 و U2 بطريقة تعتمد على لون الكرة الأولى المسحوبة من U1.

1. **Problem Statement:** We want to approximate the probability $P(X \geq x)$ for a binomial random variable $X$ using a normal distribution.

1. **State the problem:** We have a lecture hall with 190 seats, 27 of which have left-handed arm tablets. The average class size is 179 students, and 12% of these students are lef

1. **State the problem:** A box contains 10 counters, red and green, with a ratio of red to green counters as 1:4.

1. مسئله: احتمال وقوع رویداد $A$ را بیابید که مجموعه $A = \{3, \emptyset, \varnothing\}$ است. 2. ابتدا باید تعداد کل حالات ممکن را بدانیم. اگر فضای نمونه مشخص نشده باشد، نمی‌توان ا

1. **Problem Statement:** Given the probability distribution for $X$ with values $0,1,2,3$ and probabilities $0.30,0.40,0.20,$ and an unknown $P(X=3)$, find the missing probability

1. **Problem Statement:** Create the probability distribution for the experiment: "Four coins are tossed. Let X be the number of heads."

1. **Problem:** Calculate the probability that a randomly selected item from the factory is defective given the production proportions and defect rates of machines A, B, and C. 2.

1. **Problem Statement:** We have 4 equi-probable discrete input states $x_1=35$, $x_2=65$, $x_3=95$, $x_4=125$ and 3 equi-probable noise values $\eta_1=5$, $\eta_2=10$, $\eta_3=15

1. **Problem Statement:** Given the table of students by gender and major, find various probabilities related to gender and major. 2. **Given Table:**

1. **Problem:** A box contains 5 red balls, 4 blue balls, and 3 green balls. Two balls are drawn at random. Find the probability that both balls are red or both balls are blue. 2.

1. **Problem Statement:** Show that for any two events $A$ and $B$, the probability of their union is given by:

1. The problem asks for the possible values of the random variable $T$, which represents the number of tails when two coins are tossed. 2. When tossing two coins, each coin can be

1. **Problem statement:** Given probabilities in a Venn diagram with events A, B, C, D and some intersections, find mutually exclusive events, values of unknown probabilities, cond

1. **Problem statement:** We are given a joint probability density function (pdf) for random variables $X$ and $Y$:

1. **Problem Statement:** Given the joint probability density function (pdf) of random variables $X$ and $Y$:

1. **Problem Statement:** Given the joint pdf of continuous random variables $X$ and $Y$: $$f(x,y) = \frac{1}{8}(x+y), \quad 0 < x < 2, 0 < y < 2; \quad 0 \text{ otherwise}$$

1. **Problem Statement:** Calculate the covariance $\sigma_{xy}$ and correlation coefficient $\rho_{xy}$ for discrete random variables $X$ and $Y$ given their joint PMF.