🎲 probability

1. **Problem 3:** Given events A, B, and C with $$P(A) = \frac{1}{2}, \quad P(C) = \frac{1}{2}, \quad P(A \cup B) = \frac{5}{8}$$

1. **Problem Statement:** What is the probability of an impossible event?

1. **Problem Statement:** We are testing 3 cellphones and want to find the probability distribution of the random variable $x$, which represents the number of defective cellphones.

1. **Problem Statement:** We are testing 3 cellphones and want to find the probability distribution of the random variable $x$, which represents the number of defective cellphones.

1. **Problem Statement:** We have two random variables:

1. **Problem statement:** Given two independent events $A$ and $B$ in sample space $\Omega$ with $P(A \cap B) = 0.08$ and $P(A) = 2P(B)$, find $P(A \cup B)$.

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