1. **Stating the problem:** You asked for detailed information about various fundamental topics in probability theory and statistics. 2. **Random events and actions:** A random event is an outcome or set of outcomes from a random experiment. Actions on random events include union, intersection, and complement. 3. **Basic properties of probability:** Probability measures the likelihood of an event, satisfying $0 \leq P(A) \leq 1$, $P(\Omega) = 1$, and for mutually exclusive events $A$ and $B$, $P(A \cup B) = P(A) + P(B)$. 4. **Classical definition of probability:** For equally likely outcomes, $P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$. 5. **Sampling:** Repeated sampling allows replacement; non-repeated sampling does not. 6. **Combinatorics:** Basic elements include permutations, combinations, and arrangements used to count outcomes. 7. **Relative frequency and statistical probability:** Relative frequency is $\frac{\text{number of times event occurs}}{\text{total trials}}$. Statistical probability approximates true probability as trials increase. 8. **Geometric probability:** Probability based on geometric measures, e.g., lengths or areas. 9. **Conditional probability:** $P(A|B) = \frac{P(A \cap B)}{P(B)}$ if $P(B) > 0$. Events are independent if $P(A \cap B) = P(A)P(B)$. 10. **Addition and multiplication theorems:** Addition: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Multiplication: $P(A \cap B) = P(A)P(B|A)$. 11. **Total probability and Bayes' formula:** Total probability: $P(B) = \sum_i P(B|A_i)P(A_i)$ for partition $\{A_i\}$. Bayes' formula: $P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum_j P(B|A_j)P(A_j)}$. 12. **Independent trials and Bernoulli's theorem:** In $n$ independent trials with success probability $p$, the number of successes follows binomial distribution $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. Bernoulli's theorem states relative frequency converges to $p$ as $n \to \infty$. 13. **Most probable number of occurrences:** The mode of binomial distribution is approximately $\lfloor (n+1)p \rfloor$. 14. **Poisson theorem:** Approximates binomial distribution for large $n$ and small $p$ with $\lambda = np$, $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$. 15. **Moivre–Laplace theorems:** Local limit theorem approximates binomial probabilities; integral theorem approximates cumulative binomial probabilities by normal distribution. 16. **Random variables and distributions:** Discrete random variables have probability mass functions; continuous have density functions. 17. **Mathematical expectation:** $E(X) = \sum x_i P(X=x_i)$ for discrete, $E(X) = \int x f(x) dx$ for continuous variables. 18. **Variance and standard deviation:** $Var(X) = E[(X - E(X))^2]$, standard deviation is $\sqrt{Var(X)}$. 19. **Moments:** Initial moments $E(X^k)$, central moments $E[(X - E(X))^k]$ describe distribution shape. 20. **Mode, median, skewness, kurtosis:** Describe central tendency and shape of distribution. 21. **Common distributions:** Regular, exponential, normal distributions with their properties. 22. **Law of large numbers:** Sample averages converge to expected value. 23. **Markov and Chebyshev inequalities:** Provide bounds on probabilities. 24. **Statistical sampling and estimation:** Methods to estimate parameters and test hypotheses. 25. **Random processes:** Poisson, Wiener, and Markov processes model random phenomena over time. This summary covers the first question of your message, which is a broad request for detailed information on probability and statistics topics.