🎲 probability

1. **State the problem:** We have a spinning wheel with 3 green sectors, 1 blue sector, and 1 red sector. We want to find the probability of landing on a green sector and the proba

1. **State the problem:** We need to calculate the payoff of the insurance claim \( C = 5E[X^3] + 6E[X^2] \) where \(X\) is a gamma distributed random variable with parameters \(\a

1. The problem states that $X$ is an exponential random variable with rate parameter $\lambda = \frac{4}{3}$. We want to find the probability $P(X > 1)$, which is the probability t

1. **Exercice 20 : Événements contraires** L'urne contient 5 boules rouges (R) et 3 boules vertes (V). On tire 3 boules simultanément.

1. The problem is to explain why the formula $$P(success) = P(S) \times P(success \mid S) + P(\neg S) \times P(success \mid \neg S)$$ holds. 2. This formula is an application of th

1. **State the problem:** Mr. Osei wants to invest a loan of 100000 in either the phone accessories business or the mobile money business. We need to determine which business has a

1. The term "collectively exhaustive" is used in probability and set theory to describe a set of events or subsets that cover the entire sample space or universal set. 2. This mean

1. The problem is to define the term "mutually exclusive" in a clear and precise way. 2. In probability and set theory, two events are called mutually exclusive if they cannot happ

1. The problem asks to state the defining properties of a probability measure $P$ on a measurable space $(\Omega, \mathcal{F})$. 2. A probability measure $P$ is a function from the

1. **Exercice 20 : Événements contraires** L'urne contient 5 boules rouges (R) et 3 boules vertes (V). On tire 3 boules simultanément.

1. **Problem statement:** We have a binomial random variable $X$ representing the number of Canadians (out of 20) who will spend money on Halloween. The probability of success (spe

1. Problem: Find the probabilities for events A, B, and C where the sample space has 5 equally likely outcomes: E1, E2, E3, E4, E5. 2. Calculate P(A), P(B), and P(C):

1. **State the problem:** We have a continuous random variable $T$ with probability density function (pdf) $$f(t) = \begin{cases} t|\sin 2t|, & 0 \leq t \leq \pi \\ 0, & \text{othe

1. **Problem statement:** A married couple has two children. We want to find probabilities related to the gender of the children using tree diagrams. 2. **Assumptions:** Each child

1. **Define the terms:** a) **Experiment:** An experiment is a process or action that leads to one or more outcomes. Example: Tossing a coin.

1. **State the problem:** We are given a probability density function (pdf) defined as

1. **State the problem:** We have 71 guests with bookings involving breakfast (B), lunch (L), and supper (S). We need to find probabilities related to meal bookings based on the Ve

1. Problem 15: Two children are selected from a group with 10 more boys than girls. There are 756 ordered selections. Find the probability that two boys or two girls are selected.

1. **State the problem:** We have a circular archery target divided into four parts by concentric circles with radii 3 cm, 9 cm, 15 cm, and 30 cm. The scoring regions are: - Centra

1. **Problem 7:** Two fair 4-sided dice (faces 1 to 4) are rolled. Event A: sum is prime.

1. **Problem 15:** A biased pyramid-shaped die has faces 1 to 5 with probabilities given by $P(x) = \frac{k - x}{25}$ where $k$ is a constant. 2. **Find the probability of scoring