🎲 probability

1. **State the problem:** We are given a probability density function (pdf) defined as

1. **State the problem:** We have 71 guests with bookings involving breakfast (B), lunch (L), and supper (S). We need to find probabilities related to meal bookings based on the Ve

1. Problem 15: Two children are selected from a group with 10 more boys than girls. There are 756 ordered selections. Find the probability that two boys or two girls are selected.

1. **State the problem:** We have a circular archery target divided into four parts by concentric circles with radii 3 cm, 9 cm, 15 cm, and 30 cm. The scoring regions are: - Centra

1. **Problem 7:** Two fair 4-sided dice (faces 1 to 4) are rolled. Event A: sum is prime.

1. **Problem 15:** A biased pyramid-shaped die has faces 1 to 5 with probabilities given by $P(x) = \frac{k - x}{25}$ where $k$ is a constant. 2. **Find the probability of scoring

1. The problem asks for the probability of rolling a 3 on a die. 2. A standard die has 6 faces numbered from 1 to 6.

1. **Problem 14:** Given $P(A)=0.4$, $P(B)=0.7$, and $P(A \cup B)=0.8$, find: a) $P(A \cup B')$

1. Let's start by stating the problem: We want to understand why, when calculating the probability of the union of two events $A$ and $B$, we add their individual probabilities and

1. **State the problem:** We have a 30 cm by 30 cm square board with two rectangular cards attached: one 8 cm by 12 cm (Card A) and one 15 cm by 20 cm (Card B). The larger card cov

1. Problem: In an experiment involving two successive rolls of a die, you are told that the sum of the two rolls is 9. How likely is it that the first roll was a 6? Step 1: List al

1. The problem is to find the probability of selecting exactly 10 people out of 100. 2. Assuming each person is equally likely to be chosen and the selection is random without repl

1. The problem asks for the probability of getting exactly two correct answers out of 4 True or False questions by guessing. 2. Each question has 2 possible answers, so the probabi

1. The problem asks for the distribution that models the number on a card selected at random from 20 cards numbered 1 to 20. 2. Since each card is equally likely to be selected, th

1. **State the problem:** We have a random variable $X$ with pdf $$ f(x) = \begin{cases} e^{2x} & \text{if } x < 0 \\ e^{-2x} & \text{if } x \geq 0 \end{cases} $$

1. **State the problem:** We have a probability density function (pdf) $f(x)$ defined on $[0,k]$ with a triangular shape. The height at $x=0$ is $\frac{1}{2}k$ and it decreases lin

1. **Problem statement:** Transform the given sequence of random numbers uniformly distributed on [0,1] to a uniform distribution on [5,10], then find the mean of the transformed s

1. Problem 9: Given the probability density function (pdf) \( f(x) = cx^2(2-x) \) for \( 0 \leq x \leq 2 \) and 0 otherwise. (a) To find \( c \), use the property that the total pr

1. Problem 9: Given the probability density function (pdf) \( f(x) = c x^2 (2 - x) \) for \( 0 \leq x \leq 2 \) and 0 otherwise. (a) To find \( c \), use the property that the tota

1. Problem 7(a): Find the mean and variance of $X$ where $f(x) = 2(1-x)$ for $0 \leq x \leq 1$, and $0$ otherwise. 2. Calculate the mean $E(X)$:

1. Problem 4: Given the probability density function (pdf) of the lifespan $X$ of an insect: $$f(x) = \begin{cases} k \cos\left(\frac{\pi x}{1000}\right), & 0 \leq x \leq 500 \\ 0,