🎲 probability

1. **Stating the problem:** A coffee machine fills cups with an amount of coffee uniformly distributed between 130 ml and 160 ml.

1. **Problem statement:** We have a disease with prevalence $P(D) = \frac{1}{1000} = 0.001$. The test is 99% accurate, meaning $P(\text{Positive}|D) = 0.99$ (true positive rate) an

1. **Problem Statement:** Calculate the probabilities related to a standard deck of 52 playing cards.

1. **Problem Statement:** Find the probability of picking certain cards from a standard deck and answer related probability questions.

1. **State the problem:** We have a Venn diagram with sets $\mathbb{R}$, $\mathbb{T}$, and $\mathbb{V}$, and integers $a$ and $b$ representing unknown quantities in the diagram. 2.

1. **State the problem:** We have a Venn diagram with sets R, T, and V, and integers a and b representing unknown quantities in the intersections. 2. **Given probabilities:**

1. The problem provides a table with number positions and their corresponding probabilities: | संख्या स्थान (Number Position) | 7 | 9 | 8 | 10 |

1. **Problem Statement:** A basket contains 10 ripe and 4 unripe bananas. Four bananas are taken one after another. We want to find the possible values of the random variable $R$ r

1. **Problem Statement:** We have 3 cell phones tested at random, and $X$ is the random variable representing the number of defective cell phones among the 3 tested. 2. **Assumptio

1. **Stating the problem:** Define what a Markov Process is and provide an example. 2. **Definition:** A Markov Process is a type of stochastic process that satisfies the Markov pr

1. **Verify the PMF for the first distribution:** The PMF is valid if all probabilities are between 0 and 1 and sum to 1.

1. **Problem Statement:** Verify whether the given Markov chain transition matrix $$P = \begin{pmatrix} 0.3 & 0.7 & 0 \\ 0.1 & 0.4 & 0.5 \\ 0 & 0.2 & 0.8 \end{pmatrix}$$

1. **State the problem:** We want to find the probability that the second die roll is exactly one greater than the first die roll after rolling a fair six-sided die twice. 2. **Und

1. **Problem statement:** Find the probability that there are no green leaves in box A finally after the described process.

1. **State the problem:** We have a bag with only red and blue marbles. We draw three marbles and define the random variable $R$ as the number of red marbles drawn. 2. **Determine

1. **Problem Statement:** Find the total number of students in the senior class based on the Venn diagram data.

1. Problem: Find the theoretical probability of drawing certain cards from a standard 52-card deck. The formula for theoretical probability is: $$P(E) = \frac{\text{Number of favor

1. **Problem Statement:** We are given that 60% of pupils have internet access at home. From a group of 8 students chosen at random, we want to find:

1. Muammo: Ikki zavoddan keltirilgan maxsuIotlar orasida tanlangan maxsuIot sifatsiz bo'lsa, u ikkinchi zavoddan bo'lish ehtimolini topish. 2. Formulalar va tushunchalar:

1. Masala: A va B orasida uchta odam bo'lish ehtimolini topish. 2. Berilgan: A va B navbatda turibdi, ular orasida yana 8 odam bor, jami 10 odam.

1. **State the problem:** We are given a probability density function (pdf) for a random variable $X$ defined as $$f(x) = \frac{x\pi}{5m} \text{ for } 0 < x < 1, \text{ and } 0 \te