Real Numbers and Probability Concepts

Real Numbers

• Real numbers encompass all rational and irrational numbers.
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. Examples include integers (e.g., 3, -2), fractions (e.g., 1/2, 5/7), and terminating or repeating decimals (e.g., 0.5, 0.333...).
• Irrational numbers cannot be expressed as a fraction of two integers. Examples include π (pi) and the square root of 2 (√2).
• Real numbers are represented on a number line.
• Properties of real numbers include closure, commutativity, associativity, distributivity, identity elements (0 and 1), and inverse elements.

Probability

• Probability is a measure of the likelihood of an event occurring.
• It is expressed as a number between 0 and 1, inclusive.
• A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event.
• Probability can be calculated using various methods, including:
  • Theoretical probability: The ratio of favorable outcomes to the total number of possible outcomes when all outcomes are equally likely. For example, the probability of rolling a 6 on a fair six-sided die is 1/6.
  • Experimental probability: The ratio of the number of times an event occurs to the total number of trials conducted. This approach relies on data gathered from repeated experiments.
• Fundamental concepts in probability include:
  • Sample space: The set of all possible outcomes of an experiment.
  • Event: A subset of the sample space.
  • Complementary events: Two events whose union is the sample space and whose intersection is the empty set. The sum of the probabilities of complementary events is 1.
  • Mutually exclusive events: Events that cannot occur simultaneously. The probability of the union of two mutually exclusive events is the sum of their individual probabilities.
  • Independent events: Events where the occurrence of one does not affect the probability of the other occurring. The probability of two independent events both occurring is the product of their individual probabilities.
• Common probability distributions include the binomial distribution (for repeated independent Bernoulli trials), the normal distribution (a continuous probability distribution that is often used to model natural phenomena), and Poisson distribution (for the number of events that occur in a fixed interval of time or space).
• Applications of probability are widespread, including in statistics, finance, and various scientific fields.