1.4 Random Experiments and Sample Spaces

Random experiments are the foundation of statistical inference. They involve uncertain outcomes within defined sets, like flipping a coin or measuring plant growth. Understanding these experiments is crucial for grasping probability concepts.

Sample spaces represent all possible outcomes of an experiment. They can be finite (like dice rolls) or infinite (like time measurements). Knowing how to identify and work with sample spaces is key to calculating probabilities and analyzing experimental results.

Random Experiments and Sample Spaces

Components of random experiments

• Random experiment process yields uncertain outcomes within defined set (coin flip)
• Experimental unit subject of experiment (individual plant)
• Treatment specific condition applied (fertilizer type)
• Response variable measured characteristic (plant height)
• Factors influence response (soil type, sunlight)
• Repeatability allows identical conditions reproduction
• Unpredictability individual outcomes remain uncertain
• Well-defined set of possible outcomes established beforehand

Sample spaces in experiments

• Sample space set of all possible outcomes denoted S or Ω
• List method enumerates outcomes (coin toss: H, T)
• Tree diagram visually represents sequential events (two coin tosses)
• Cartesian product for multi-step experiments (dice roll and coin toss)
• Finite sample spaces limited outcomes (die roll: 1-6)
• Infinite sample spaces unlimited outcomes (time measurement)

Simple vs compound events

• Simple event single indivisible outcome (rolling a 6)
• Compound event combination of simple events (rolling even number)
• Simple events mutually exclusive
• Compound events may overlap
• Notation uses lowercase for simple (a, b, c) uppercase for compound (A, B, C)

Probability calculation methods

• Probability measures likelihood ranges 0 to 1
• Sample space calculation: favorable outcomes / total outcomes
• Multiplication Rule independent events: $P(A \text{ and } B) = P(A) \times P(B)$
• Dependent events: $P(A \text{ and } B) = P(A) \times P(B|A)$
• Complementary events: $P(A') = 1 - P(A)$
• Addition Rule mutually exclusive: $P(A \text{ or } B) = P(A) + P(B)$
• Non-mutually exclusive: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$