1.1 Basic concepts of probability and randomness

Probability fundamentals provide the foundation for understanding uncertainty in engineering. We'll explore how probability quantifies event likelihood, distinguishes between deterministic and random phenomena, and defines random experiments and their outcomes.

Relative frequency plays a crucial role in connecting theoretical probability to real-world observations. We'll examine how it estimates probabilities and relates to the law of large numbers, essential concepts for engineers analyzing data and making predictions.

Probability Fundamentals

Definition and role of probability

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• Probability numerically measures the likelihood of an event occurring
  • Assigns a value between 0 (impossible event) and 1 (certain event) to quantify the likelihood
  • Provides a consistent framework to quantify and analyze uncertainty
• Probability enables informed decision-making under uncertainty
  • Assesses risks and predicts future outcomes (weather forecasting, financial investments)
  • Develops models for complex systems with random components (stock market, traffic flow)

Deterministic vs random phenomena

• Deterministic phenomena have predictable outcomes
  • Same initial conditions always yield the same result
  • Examples: simple machines (levers, pulleys), idealized physical systems (frictionless surfaces)
• Random phenomena have unpredictable outcomes
  • Outcome varies even with identical initial conditions
  • Examples: rolling a die, flipping a coin, weather patterns
• Probability describes and analyzes random phenomena
  • Deterministic phenomena do not require probabilistic analysis

Random experiments and outcomes

• Random experiment generates an outcome that cannot be predicted with certainty
  • Sample space is the set of all possible outcomes
• Examples of random experiments and their outcomes:
  • Tossing a coin (heads, tails)
  • Rolling a six-sided die (1, 2, 3, 4, 5, 6)
  • Drawing a card from a well-shuffled deck (any of the 52 cards)
  • Measuring time between customer arrivals at a store (continuous values)

Relative frequency in probability

• Relative frequency is the ratio of the number of times an event occurs to the total number of trials
  • $Relative frequency = \frac{Number of times the event occurs}{Total number of trials}$
• As the number of trials increases, the relative frequency of an event stabilizes around a value
  • This stable value estimates the probability of the event
• The law of large numbers states that relative frequency converges to probability as trials approach infinity
  • Links theoretical probability to empirical observation of relative frequency
  • Example: flipping a fair coin many times, relative frequency of heads approaches 0.5