🎲 probability

1. **State the problem:** We have a bag with red and blue counters in the ratio 4 : 10. 2. **What is asked:** Find the probability that a randomly chosen counter is blue.

1. **Problem 1: Given the joint probability distribution of X and Y as** $$\begin{array}{c|ccc}

1. **Stating the problem:** We have a probability table for the outcomes of a game: win, lose, and draw. The probabilities for win and lose are given as 0.3 and 0.25 respectively,

1. **Problem 1: Lottery Tickets** We have 5000 tickets sold at 1 each.

1. **State the problem:** We have three bags A, B, and C with red and white counters. Ratios and counts are given, and counters are moved between bags. We want the probability that

1. **Problem:** Find the theoretical probability of selecting a number from the sample space \{2, 3, 4, 5\}. Theoretical probability formula: $$P(E) = \frac{\text{Number of favorab

1. **Problem Statement:** We have a Markov chain with two states A and B and transition matrix $$P = \begin{bmatrix}0.9 & 0.1 \\ 0.3 & 0.7\end{bmatrix}$$. We want to find the long-

1. **Problem Statement:** We have an intersection with an average of 150 cars passing through every hour. We want to analyze the number of cars passing through in a 1-minute period

1. **Problem Statement:** We have five cards numbered 1 through 5. Three cards are randomly flipped over. We want to find the probability that the sum of the numbers on these three

1. **Problem statement:** We have a binomial random variable $X$ with parameters $n=30$ and $p=0.60$. We want to find using the normal approximation: (i) $P(X=14)$ (the probability

1. **Problem statement:** We are given a binomial distribution with parameters $n=100$ and $p=0.2$ (probability of defective chip). We want to find the probabilities for (i) at mos

1. **Problem Statement:** Find the probability of picking two kings when selecting 2 cards from a standard 52-card deck. 2. **Formula:** The probability of an event is given by

1. **Problem statement:** We have defects occurring along a cable following a Poisson process with rate $\lambda = 0.2$ defects per kilometre. Given no defects in the first 2 kilom

1. **Problem Statement:** We have a drug with a survival rate of 70% (probability of survival $p=0.7$) and death rate $q=1-p=0.3$. For 12 women taking the drug, we want to find:

1. **Problem Statement:** We have 3 coin tosses with outcomes either W (Worn attack) or N (None). We want to analyze the random variable $X$ which counts the number of tails (W) in

1. **Stating the problem:** We have 6 Democrats and 4 Republicans from the same city in the House of Representatives. We want to find the probability that all four leadership posit

1. **Problem Statement:** We want to find the probability of getting a specific license plate "ABC012" where the plate format is three letters (no duplicates) followed by three num

1. **Problem Statement:** We have a group of 160 volunteers. Define sets: - $S = \{\text{all 160 people}\}$

1. State the problem. We are given the probability distribution of the daily demand X for a movie magazine with values 0 through 9 and their probabilities as follows.

1. **Stating the problem:** A couple has five sons, and we want to find the probability that their next baby will be a daughter. 2. **Understanding the problem:** The gender of eac

1. **Problem Statement:** We have two special 4-sided dice, each with faces numbered 1 to 4. We roll both dice and define two random variables: $X = \min(D_1, D_2)$ and $Y = \max(D