🎲 probability

1. The problem asks for the probability of heads appearing on the 5th toss of a fair coin, given that tails appeared on the first four tosses. 2. A fair coin means the probability

1. Stating the problem: Let's explore some fundamental probability examples to understand basic concepts. 2. Example 1: Probability of drawing an ace from a standard deck of 52 car

1. Stating the problem: We are given the probabilities of choosing socks of different colors: - $P(\text{pink}) = \frac{1}{8}$

1. **Determine the missing probability in the distribution.** Given probabilities: $0.07, 0.20, 0.38, ?, 0.13$

1. Let's state the first problem: We have four inspectors who stamp film packages with different probabilities of failure. We want the probability that a package missing the expira

1. Define Discrete Probability. Give an example. Discrete probability deals with outcomes that can be counted and listed. Each outcome has a probability, and the sum of all probabi

1. Énoncer le problème : Nous avons un tirage successif avec remise de trois boules dans une boîte (Exercice 1) et un tirage simultané puis successif sans remise dans une urne avec

1. **State the problem:** We are given a random variable $X$ with probability density function (pdf) $$f(x)=\frac{4x(9-x)}{81},\quad 0 \leq x \leq 3.$$ We need to find the mean dev

1. Let us state the problem: We have a multiple choice test with 20 questions, each having 4 possible answers (a,b,c,d), only one of which is correct. 2. We want the probability of

1. **Problem Statement:** There are 16 tennis balls in total, 10 of which have never been used. Two balls are chosen and played with, then returned to the box. Later, two balls are

1. **State the problem:** There are 16 tennis balls in total, 10 of which are new (not previously used). Two balls are chosen, played with, and put back. Then, two balls are chosen

1. Problem Statement: (a) State two conditions required for a discrete random variable $X$ to be modelled by a binomial distribution.

**Problem 5:** Given a probability space [\(\Omega, \mathcal{A}, P\)] and events \(A, B \in \mathcal{A}\) with \(P(A \cap B)=0.26\), \(P(A \cup B)=0.80\), and \(P(\overline{B})=0.3

1. **Multiplication Law of Probability:** The multiplication law states that for two events A and B, the probability of both occurring is:

1. **Understanding the problem:** We have mutually exclusive events $A$ and $B$ in the sample space $2^\Omega$. We want to find the probability of $\overline{A} \cap \overline{B}$,

1. **Problem statement:** Given the piecewise probability density function $$f(x) = \begin{cases} c(x + 3) & 0 < x < 2 \\ c(7 - x) & 2 < x < 4 \\ 0 & \text{elsewhere} \end{cases}$$

1. **Problem Statement:** Calculate the expectation $E[Z_2]$ for a Galton-Watson branching process with offspring generating function

1. **State the problem:** Find the constant $c$ for the given PDF $f(x) = c x e^{-2x}, x > 0$, with total probability 1.

1. **Stating the problem:** Find the constant $c$ such that $f(x) = c x e^{-2x}, x > 0$ is a valid pdf. 2. **Total probability must be 1:**

1. **Problem statement:** Find the constant $c$ given the probability density function (pdf) $f(x) = c x e^{-2x}, x > 0$. The total probability must integrate to 1: $$\int_0^{\inft

1. **State the problem:** We need to find the probability that the spinner lands on section D given the probabilities for sections A, B, and C. 2. **List the given probabilities:**