🎲 probability

1. **State the problem:** We have a random variable $X$ with probability mass function (pmf) $f(x) = \frac{1}{6}$ for $x = 1, 2, 3, 4, 5, 6$ and $0$ otherwise. We define a new rand

1. **State the problem:** We have a fleet of planes with probabilities: no wing cracks $=0.7$, detectable wing cracks $=0.25$, and critical wing cracks $=0.05$. For the next four p

1. **Problem Statement:** We have two dice: the first with faces 1 to 6, the second with faces 1 to 10. Let $X$ be the sum of the numbers rolled on both dice. We want to find the v

1. **State the problem:** We need to find the value of $$R + \mathrm{Var}(Y) + P(Y < 2)$$ where:

1. **State the problem:** We are given the generating function $$G_X(s) = \frac{s^3}{6} (2 + s^4 + 2s^8)$$ and asked to find $$\text{mean}(3X + 5) + \text{Var}(5X + 7).$$

1. The problem is to understand the meaning of "iid" in probability and statistics. 2. "iid" stands for "independent and identically distributed". This means that a set of random v

1. **Problem statement:** Given continuous random variables $(X,Y)$ with joint density function $$f_{X,Y}(x,y) = k y e^{-y(1+x)}, \quad x>0, y>0,$$ and zero otherwise, where $k>0$,

1. The problem asks whether the probability of rolling a five on a six-sided die is about 17%. 2. The formula for the probability of a single outcome on a fair six-sided die is

1. **State the problem:** Given probabilities $P(A) = \frac{1}{2}$, $P(B) = \frac{2}{3}$, $P(A \cap B) = \frac{1}{4}$, and $P(A \cup B) = \frac{1}{6}$, determine if events $A$ and

1. **State the problem:** We want to determine if event A (student is female) and event B (student prefers romance movies) are independent. 2. **Recall the formula for independence

1. **State the problem:** We want to find the probability of drawing an ace or a jack from a standard deck of 52 cards. 2. **Formula:** The probability of an event is given by

1. **State the problem:** You flip a coin twice. The probability of heads on any flip is $\frac{1}{2}$. Given the first flip is heads, find the probability the second flip is tails

1. The problem asks for the number of possible outcomes when rolling a fair six-sided die. 2. A six-sided die has faces numbered from 1 to 6.

1. **State the problem:** We want to find the probability $P[A]$ of rolling a total of 2, 3, 4, or 5 with two dice. 2. **Sample space:** The total number of possible outcomes when

1. **Problem statement:** Find the probability of selecting various cards from a shuffled 52-card deck. 2. **Formula for probability:**

1. The problem asks: What is an event consisting of one outcome called? 2. The formula or concept: An event with exactly one outcome is called a simple event.

1. **Problem:** Find the probability that the secretary is female and the chairman and treasurer are of different genders. 2. **Given:** 15 males, 10 females, total 25 people.

1. **State the problem:** Given the probability of event $A$ as $p(A) = 0.4$, find the probability of the complement of $A$, denoted as $A^c$. 2. **Formula used:** The probability

1. **Stating the problem:** We roll a fair six-sided die and define event $E$ as getting an even number. 2. **Understanding the sample space:** The sample space $S$ for a six-sided

1. **State the problem:** Find the variance and standard deviation of a given probability distribution with values $x$ and probabilities $P(X)$. 2. **Recall formulas:**

1. **Problem Statement:** Find the mean and variance of a Poisson distribution with parameter $\lambda$. 2. **Recall the probability generating function (p.g.f) of a Poisson distri