🎲 probability

1. **Problem Statement:** Given the joint probability mass function (pmf) of random variables $X$ and $Y$:

1. **Problem Statement:** Given the joint probability density function (pdf) for $(X_1, X_2)$ with $-1 \leq \alpha \leq 1$:

1. **Problem Statement:** Given the joint pdf $$f(x,y) = \frac{2}{(1+x+y)^3}$$ for $$x>0, y>0$$ and 0 elsewhere, find: (i) Marginal densities of $$X$$ and $$Y$$

1. **Problem statement:** Given the joint probability density function (pdf) of $X$ and $Y$: $$p(x,y) = \begin{cases} k(8 - x - y), & 0 < x < 3, 0 < y < 4 \\ 0, & \text{otherwise}

1. **Problem Statement:** We have a joint probability density function (p.d.f) of random variables $X$ and $Y$ given by:

1. **Problem Statement:** (a) Given joint pdf $f_{X,Y}(x,y) = \frac{15}{2}(2 - x - y)$ for $0 < x < 1$, $0 < y < 1$, find the conditional density $f_{X|Y}(x|y)$.

1. **Problem statement:** Given the joint pdf $$f_{X,Y}(x,y) = \frac{15}{2}(2 - x - y)$$ for $$0 < x < 1$$ and $$0 < y < 1$$, find the conditional density of $$X$$ given $$Y = y$$.

1. **Problem Statement:** (a) Given joint pdf $f_{X,Y}(x,y) = \frac{15}{2}(2 - x - y)$ for $0 < x < 1$, $0 < y < 1$, find the conditional density of $X$ given $Y = y$.

1. **Problem statement:** We roll a fair six-sided die repeatedly until we get a 1 or a 6. We want to find the probability that it takes at least 3 throws but no more than 5 throws

1. **Problem statement:** We roll a fair six-sided die repeatedly until we get a 1 or a 6. We want to find the probability that it takes at least 3 throws but no more than 5 throws

1. **Problem Statement:** A discrete random variable $X$ has a probability mass function (pmf) given by:

1. **Problem statement:** A discrete random variable $X$ has a probability mass function (pmf) given by $f(x) = c(x + 1)$ for $x = 0, 1, 2, 3$. We need to find the constant $c$. 2.

1. Problem: Calculate the theoretical probability of getting an odd number when rolling a fair dice. 2. Formula: The theoretical probability of an event is given by

1. Problem: Calculate the theoretical probability of getting an odd number when rolling a fair dice. 2. Formula: The theoretical probability of an event $E$ is given by

1. The problem is to understand and calculate the conditional probabilities $P(X_2|C_1)$ and $P(X_2|C_2)$. 2. Conditional probability $P(A|B)$ is defined as the probability of even

1. **Problem Statement:** They can play a maximum of 8 games and want to guarantee winning the prize. We need to find the minimum number of board games they should play to ensure t

1. **Problem Statement:** We have a roulette wheel with numbers 1 to 9 and their respective probabilities given as: | X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

1. The problem is to understand and use Venn diagrams and tree diagrams to represent sets and probabilities. 2. A Venn diagram visually shows the relationships between different se

1. **Stating the problem:** Define the following probability terms based on the context of rolling dice and flipping coins: Experiment, Sample Space, Event, Outcome, Probability, M

1. **Problem Statement:** We need to find the probability of being allocated certain types of seats on an airplane given the layout and which seats are occupied. 2. **Understanding

1. **Problem Statement:** You are given a seating layout on an aeroplane with some seats occupied (shaded) and some unoccupied. You are to find the probability of being allocated a