🎲 probability

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,

1. Problem 4: Given the probability density function (pdf) of lifespan $f(x) = k \cos\left(\frac{\pi x}{1000}\right)$ for $0 \leq x \leq 500$, and 0 otherwise. (a) Show that $k = \

1. **Problem statement:** (a)(i) In a class of 40 students, 20 take Science, 10 take Chemistry, and 5 take both Science and Chemistry. Draw a Venn diagram to represent this.

1. The problem asks for the expected number of faulty electronic components in a box of 50, given the probability of a component being faulty is 4%. 2. The probability of a compone

1. The problem asks for the probability of drawing a black 7 from a standard deck of 52 cards. 2. A standard deck has 52 cards: 26 black cards (spades and clubs) and 26 red cards (

1. Problem statement: We have a class of 40 students with 20 taking Science, 17 taking Commerce, and 7 taking both.

1. Problem: Given $X \sim \text{Poisson}(50)$, use the normal approximation to find: a) $P(52 < X < 56)$