🎲 probability

1. **Stating the problem:** We have a bag with counters of four colors: pink, yellow, green, and blue. We want to understand the probabilities of drawing each color at random from

1. **Problem statement:** There are red and blue counters in a bag with a ratio of red to blue counters as 3:1. Two counters are removed at random, and the probability that both ar

1. **הבעיה:** להוכיח כי עבור משתנים מקריים X,Y עם מומנטים שניים סופיים, ופונקציה הפיכה $g:\mathbb{R} \to \mathbb{R}$ מתקיים: $$E[X|g(Y)] = E[X|Y]$$

1. **Problem statement:** A meeting has 4 Americans and 2 Germans. Three consuls are selected at random. We want to find the probability distribution of the random variable $G$, wh

1. **Problem statement:** A bag contains five marbles: 1 blue (B), 1 red (R), and 3 green (G). Two marbles are selected without replacement. We need to list all possible outcomes u

1. **Problem Statement:** Illustrate the sample space using a tree diagram for the orderings in which 3 girls Sarah, Tracy, and Beth may sit in a row of 3 chairs.

1. **State the problem:** Jessica has a probability of $\frac{4}{5}$ of getting an A in Mathematics and a probability of $\frac{2}{5}$ of getting an A in English. We want to find t

1. **Problem Statement:** We have a color wheel with four colors: red, orange, yellow, and green. The wheel is spun twice, selecting two colors. Define the random variable $X$ as t

1. **Problem Statement:** We want to approximate a binomial distribution $S_n \sim \text{Bin}(n, p)$ using a normal distribution when $n$ is large.

1. **Problem statement:** We have 77 female and 77 male applicants (total 154) for 5 positions. We want the probability of selecting exactly 3 females and 2 males when choosing 5 a

1. **Problem statement:** We have 1212 distinct brands of beer, and we randomly choose 33 distinct brands without repetition. We want to find the number of ways to choose these 33

1. **Problem Statement:** Find the theoretical probability of the spinner landing on red when spun once. 2. **Formula:** The theoretical probability of an event is given by:

1. **Problem:** Find $P(Y \cap Z)$ given $P(U \cup Z) = \frac{2}{3}$, $P(Y) = \frac{2}{9}$, and $P(Z) = \frac{1}{2}$. 2. **Recall the formula:** For any two events $A$ and $B$,

1. **Problem:** Find $P(Y \cap Z)$ given $P(Y \cup Z) = \frac{2}{3}$, $P(Y) = \frac{2}{9}$, and $P(Z) = \frac{1}{2}$. 2. **Formula:** Use the formula for the union of two events:

1. **Problem statement:** Show that for integer $k \geq 1$, $$\int_\mu^\infty \frac{1}{\Gamma(k)} z^{k-1} e^{-z} \, dz = \sum_{x=0}^{k-1} \frac{\mu^x e^{-\mu}}{x!}$$

1. The problem asks us to understand the sample space of a tennis match where John and Peter play sets, and the first to win two sets wins the match. 2. The sample space lists all

1. The problem involves calculating the probability of drawing marbles from two bags, X and Y, with given probabilities for each draw. 2. The first step is to understand the tree d

1. **State the problem:** We have two bags, X and Y, with different compositions of white and red marbles. A bag is chosen at random, and two marbles are drawn without replacement.

1. **Problem Statement:** A bag contains 6 red marbles and 5 black marbles, total 11 marbles. Two marbles are drawn without replacement. 2. **Part A: Construct a Probability Tree**

1. **Problem Statement:** We are given the cumulative distribution function (CDF) $F(x)$ of a uniform random variable $X \sim \text{Unif}(a,b)$.

1. **Problem (a)(i): Find the probability that both students chosen are retakers.** - Total students = 10