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Basic Probability Concepts and Rules

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  • What is an experiment in probability?

    An experiment is an activity or procedure that generates data or outcomes, such as tossing a coin or recording ages of a group.

  • Define an event in probability.

    An event is any collection of results or outcomes from an experiment.

  • What is a simple event?

    A simple event is an outcome that cannot be broken down further into simpler components.

  • What is a compound event?

    A compound event includes two or more simple events combined.

  • What is the sample space?

    The sample space is the set of all possible outcomes of an experiment.

  • Explain the concept of subset (subevent) in probability.

    A set A is a subset of B (A βŠ† B) if every member of A is also in B, meaning occurrence of A implies occurrence of B.

  • What is the complement of an event A?

    The complement of event A, denoted 𝐴̅ or 𝐴′, includes all outcomes in the sample space not in A.

  • Define the union of two events A and B.

    The union (A βˆͺ B) is the event containing all outcomes in A, or B, or both.

  • Define the intersection of two events A and B.

    The intersection (A ∩ B) is the event containing outcomes common to both A and B.

  • What are mutually exclusive (disjoint) events?

    Mutually exclusive or disjoint events have no common outcomes.

  • What is a contingency table?

    A contingency table is a two-way frequency table used to display categorical data counts.

  • State the basic properties of probability.

    1. Probability is between 0 and 1.
    2. Probability of the sample space is 1.
    3. Probability of an impossible event is 0.

  • How is probability approximated by relative frequency?

    Probability of event A β‰ˆ (number of times A occurs) / (number of trials).

  • What is the classical approach to probability?

    When outcomes are equally likely, \(P(A) = \frac{\text{number of outcomes in A}}{\text{total number of outcomes}}\).

  • What is subjective probability?

    Probability estimated based on personal judgment or knowledge rather than exact calculation.

  • State the addition rule for probabilities.

    \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), the probability of A or B occurring.

  • What is the multiplication rule for probabilities?

    \(P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)\), the probability of both A and B occurring.

  • When are two events independent?

    Events A and B are independent if \(P(B|A) = P(B)\) or equivalently \(P(A \cap B) = P(A) \cdot P(B)\).

  • Difference between mutually exclusive and independent events?

    Mutually exclusive events cannot occur together; independent events' occurrence does not affect each other.

  • What is sampling with replacement?

    Sampling where each selected item is returned before the next selection; selections are independent.

  • What is sampling without replacement?

    Sampling where selected items are not returned; selections are dependent.

  • State the complementation rule in probability.

    \(P(A) = 1 - P(\overline{A})\), the probability of event A equals one minus the probability of its complement.

  • Define conditional probability.

    Probability of A given B has occurred: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\).

  • How to find probability of event A given event B?

    Use \(P(A|B) = \frac{P(A \cap B)}{P(B)}\), considering the reduced sample space of B.

  • What is the Law of Large Numbers?

    As the number of trials increases, the relative frequency of an event approaches its true probability.