🎲 probability

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) $P(52 < X < 56)$

1. **Problem statement:** We have a class of 40 students with 25 taking Biology, 18 taking Chemistry, and 8 taking both subjects.

1. The two faces of a coin are called **Heads** and **Tails**. 2. A die has **6 faces**.

1. **State the problem:** We have a bag with 10 gum balls, 7 candy bars, and 3 toffees, totaling $10 + 7 + 3 = 20$ candies. Two candies are drawn without replacement. 2. **Part a)

1. **State the problem:** There are 180 people in a competition. The probability of winning for each person is $\frac{1}{6}$.

1. **State the problem:** Anna has 2 coins totaling 30p, and Tom has 4 coins totaling 30p. We want to find who is more likely to pick a 10p coin from their respective bags.

1. **State the problem:** We have a spinner divided into 10 equal sections, with 2 sections shaded pink. We want to find:

1. Problem: A bag contains 1 red ball, 2 green balls, and 4 yellow balls. A ball is drawn, replaced, and a second ball is drawn. Find probabilities for: (a) both balls are the same

1. **Problem:** Show that the expected value of a constant $k$ is $k$. Step 1: By definition, the expected value $E(k)$ is the sum over all outcomes of $k$ times their probabilitie

1. **State the problem:** We have a continuous random variable $T$ representing the lifetime of a battery with probability density function (pdf) $$f(t) = \begin{cases} ce^{-3t}, &

1. The problem asks for the probability that in a sample of 25 tablets, two or more are defective, given a 5% defect rate per tablet. 2. Let $X$ be the number of defective tablets

1. The problem states that Brian wants to insure a painting against theft with a probability of theft $p = 0.02$ during the year. 2. The value of the painting (or the insurance cov

1. The problem states that Brian wants to insure a painting against theft with a probability of theft $p = 0.02$ during the year. 2. The insurance policy covers a loss of $10,000 i

1. The moment generating function (MGF) of a random variable $X$ is defined as $$M_X(t) = E[e^{tX}]$$ where $E$ denotes the expected value and $t$ is a real number. 2. The MGF, if

1. **State the problem:** We want to find the maximum value of $p$ such that the probability of no faulty chips $P(\text{no fault})$ is at least 0.92, given the approximate express

1. **Problem statement:** We have 100 letters and 100 envelopes, and letters are randomly inserted into envelopes. We want to find the probabilities for different numbers of letter