🎲 probability

1. The problem is to explain key terms under probability. 2. Probability is the measure of how likely an event is to occur, expressed as a number between 0 and 1.

1. **Problem Statement:** Calculate the probability of drawing a specific card from a standard deck of 52 cards. 2. **Formula:** Probability of an event = \frac{Number of favorable

1. Let's start by understanding what probability is. Probability measures how likely an event is to happen. It is a number between 0 and 1, where 0 means the event will not happen

1. **State the problem:** We have a fruit bowl with oranges, kiwis, and limes. The ratio of oranges to kiwis is 3:5.

1. Problem: A fair coin is tossed 3 times. Find the probability that at least one head appears. 2. Formula: The probability of at least one head is the complement of the probabilit

1. The problem is to find the probability of "at least one event" occurring in a given set of events. 2. The formula to calculate the probability of at least one event happening is

1. The problem asks: If two events are collectively exhaustive, what is the probability that one or the other occurs? 2. **Definition:** Two events are collectively exhaustive if a

1. **Problem Statement:** Two fair dice are thrown simultaneously. Let the random variable $X$ be the sum of the two outcomes. We need to find the probabilities for each possible s

1. **State the problem:** Martin has 5 plain and 3 stripy socks, total 8 socks. He picks one sock at random, then picks a second sock at random without replacement. We want to find

1. **State the problem:** We have 11 animals: 7 bats and 4 mice. One animal leaves at random, then a second animal leaves at random without replacement. We want to complete the tre

1. **State the problem:** Jim has 9 pieces of fruit: 7 grapefruits and 2 apples. He picks one fruit at random and eats it, then picks a second fruit at random. We want to find the

1. **State the problem:** Tina has two bags, A and B, with red and blue counters. We want to complete the probability tree diagram showing the probabilities of drawing red or blue

1. **State the problem:** We need to find the probability that Arun scores the winning goal on a Saturday. 2. **Given information:**

1. **Problem a:** Show that for a random vector $\mathbf{X}$, $\mathrm{Var}(A\mathbf{X} + \mathbf{b}) = A \mathrm{Var}(\mathbf{X}) A^T$ where $A$ is a $(k \times p)$ matrix and $\m

1. **Problem statement:** A family has three children, and it is known that at least one of the children is a girl. We want to find the probability that all three children are girl

1. **State the problem:** We have a random variable $X$ uniformly distributed on the interval $(-3,5)$, and another random variable $Y = e^{-X/3}$. We want to find the expected val

1. **State the problem:** We want to find the probability that Bob plays football on exactly 2 out of the next 3 Saturdays. 2. **Identify the type of problem:** This is a binomial

1. The problem asks for the probability of picking a joker from a standard deck of cards. 2. A standard deck of cards typically contains 52 cards plus 2 jokers, making a total of 5

1. The problem asks for the probability of picking a joker from a standard deck of cards. 2. A standard deck of cards typically contains 52 cards plus 2 jokers, making a total of 5

1. **State the problem:** There are 50 students, and 15 are chosen at random. We want the probability that you or your friend (or both) are chosen. 2. **Formula and rules:** The to

1. **Problem statement:** There are 50 students in a class, and the professor chooses 15 students at random. We want to find the probability that you or your friend (or both) are a