🎲 probability

1. The user requested a tree diagram, which is a visual representation used to show all possible outcomes of an event or series of events. 2. Tree diagrams are not mathematical pro

1. **Stating the problem:** We have a survey of 1000 students at University ABC, with 750 boys and 250 girls. Among smokers, 22.9% are boys and 4.5% are girls. We need to construct

1. **Problem:** Given the pdf \(f(x) = 0.075x + 0.2\) for \(3 \leq x \leq 5\) and 0 otherwise, verify it is a valid pdf and calculate probabilities. 2. **Step 1: Verify total area

1. **Problem:** Given the probability density function (pdf) $$f(x) = \begin{cases} 0.075x + 0.2 & 3 \leq x \leq 5 \\ 0 & \text{otherwise} \end{cases}$$

1. **Problem statement:** A committee consists of 7 men and 4 women. A sub-committee of 6 is chosen at random. Find the probability that the sub-committee contains exactly 2 women.

1. **Problem Statement:** An experiment consists of asking 3 persons at random if they wash their clothes using detergent. (a) List the elements of a sample space $S$ using the let

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