Probability and set theory

• What does probability theory help to describe?

  Situations with imperfect knowledge of factors
• Why is probabilistic reasoning important in modern life?

  It helps follow debates in economics and finance
• What is a random experiment?

  A process leading to distinct possible results
• What is the outcome of a random experiment?

  One distinct possible result of an experiment
• What is the sample space in probability?

  Set of all possible outcomes of an experiment
• How is the sample space denoted?

  By the symbol ��
• What does the sample space for rolling a six-sided die consist of?

  Numbers 𝑆 = {1, 2, 3, 4, 5, 6}
• What is an event in probability?

  A collection of one or more outcomes of an experiment
• What does the complement of an event A represent?

  Outcomes in 𝑆 that do not belong to A
• How is the complement of event A denoted?

  By 𝐴̅
• What is the null set in probability?

  The complement of the sample space, symbolized as ∅
• What does set inclusion (⊂) indicate?

  Event B is contained in event A
• What does the union of two events A and B represent?

  All outcomes that belong to A or B or both
• How is the union of events A and B denoted?

  By C = 𝐴 ∪ 𝐵
• What does the intersection of two events A and B represent?

  All outcomes that belong to both A and B
• How is the intersection of events A and B denoted?

  By C = 𝐴 ∩ 𝐵
• What are mutually exclusive events?

  Events with no outcomes in common
• What is the symbol for the null set?

  ∅
• What does it mean for events to be collectively exhaustive?

  Their union contains the entire sample space
• What is the relationship between the sample space and collectively exhaustive events?

  Every outcome of the random experiment is in S
• What does it mean for events A and C to be mutually exclusive?

  They cannot occur at the same time.
• What does the darkened region in the Venn diagram represent?

  It indicates outcomes belonging to both A and B.
• What is the intersection of an even and odd throw of a die?

  The empty set.
• What does it mean for events to be collectively exhaustive?

  Their union contains the entire sample space.
• What is a partition of the sample space?

  Disjoint events that are also exhaustive.
• What is the formula for classical probability?

  $P(A) =$$\frac{N_A}{N}$
• How do you calculate the probability of throwing a die less than 4?

  $P(A) =$$\frac{3}{6} =$$\frac{1}{2}$
• What does the first probability postulate state?

  Probability is a nonnegative real number.
• What does the second probability postulate express?

  Probability of an event is the sum of outcomes.
• What does the third probability postulate imply?

  The total probability of all events is 1.
• What is the empirical probability approach?

  It calculates probabilities based on observed frequencies.
• How is the empirical probability of flipping a coin calculated?

  $P(H) =$$\frac{45}{100} =$$\frac{9}{20}$
• Why might classical probability not be reliable in some cases?

  Outcomes may not be equally likely.
• What is subjective probability?

  Probability based on personal beliefs.
• How can subjective probabilities be derived?

  From personal beliefs or data analysis.
• What motivated the man's wager in the Scottish referendum?

  His strong confidence in the outcome.
• What is subjective probability based on?

  Personal beliefs and information
• How can subjective probability differ among individuals?

  It diverges based on differing beliefs
• What can subjective probability stem from?

  Personal beliefs, data, or other information
• What does the probability of an event represent according to the text?

  Relative frequency of occurrence over repetitions