🎲 probability

1. **Problem statement:** Given the piecewise probability density function $$f(x) = \begin{cases} c(x + 3) & 0 < x < 2 \\ c(7 - x) & 2 < x < 4 \\ 0 & \text{elsewhere} \end{cases}$$

1. **Problem Statement:** Calculate the expectation $E[Z_2]$ for a Galton-Watson branching process with offspring generating function

1. **State the problem:** Find the constant $c$ for the given PDF $f(x) = c x e^{-2x}, x > 0$, with total probability 1.

1. **Stating the problem:** Find the constant $c$ such that $f(x) = c x e^{-2x}, x > 0$ is a valid pdf. 2. **Total probability must be 1:**

1. **Problem statement:** Find the constant $c$ given the probability density function (pdf) $f(x) = c x e^{-2x}, x > 0$. The total probability must integrate to 1: $$\int_0^{\inft

1. **State the problem:** We need to find the probability that the spinner lands on section D given the probabilities for sections A, B, and C. 2. **List the given probabilities:**

1. The problem asks for the probability that a student chosen at random has either brown hair or ginger hair. 2. We are given:

1. **Problem statement:** We have a pack of 10 watches, of which 3 are defective. We select 2 watches at random. We want to find the probability that at least one of the selected w

1. **State the problem:** We want to find two events from the list that are mutually exclusive when rolling a fair six-sided die once. 2. **Define the events:**

1. **Problem Statement:** The probability density function (pdf) $f(x)$ is defined as: $$f(x) = \begin{cases} c(x + 3), & 0 < x < 2 \\ c(7 - x), & 2 < x < 4 \\ 0, & \text{elsewhere

1. The problem is to find the Moment Generating Function (MGF) for a random variable $X$ with probabilities $P(X=k) = \frac{1}{k}$ for $k=1,2,3,\ldots$ using the geometric series.

1. The problem is to find the moment generating function (MGF) of a random variable $X$ that takes values $1/k$ for $k=1,2,3,\ldots$ using the geometric series.\n\n2. Assume the PM

1. The problem asks us to find the Moment Generating Function (MGF) for a random variable defined as $X = \frac{1}{k}$, where $k = 1,2,3,\ldots$. 2. Typically, the MGF of a random

1. For the lunch and gift shop problem, state given info: - Students brought packed lunch: 43

1. **State the problem:** We have a probability $p=0.008$ that a new car has faulty brakes. A sample of $n=520$ cars is taken, and $X$ denotes the number of cars with faulty brakes

1. **Problem Statement:** Prove the Law of Total Probability: If $A_1, A_2, \dots, A_n$ are mutually exclusive and exhaustive events with $P(A_i) > 0$ for all $i=1,2,\dots,n$, and

1. **Problem Statement:** Prove the Law of Total Probability which states that if $A_1, A_2, ..., A_n$ are mutually exclusive and exhaustive events with $P(A_i) > 0$ for $i=1,2,...

1. The problem asks to find probabilities of numbers of bacteria in 1 mg of liquid, given that the mean number (\(\lambda\)) is 2 (assumed from provided probabilities). We use the

1. **State the problem:** Given the probability of a bad reaction from serum injection is 0.001, and 2000 individuals are injected, find: (i) The probability exactly 3 individuals

1. **State the problem:** Find the probability that a randomly selected student wears a watch from the group of 10 students. 2. **Identify students who wear a watch:** These are al

1. **ප්‍රශ්නය ප්‍රකාශය**: යබෝල 15ක් ඇත, ඒ අතර රතු 3, කහ 4, යකාළ 3 සහ නිල් 5 යබෝල වේ. යතාරැවින් යබෝල 3ක් තෝරා ගනී. 2. **නියැදි අවකාශය (Sample space):**