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Statistics for Business - Probability and Counting

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  • What is the theoretical probability of an event?

    Theoretical probability is the probability based on what could happen, calculated before events occur.

  • What is empirical (experimental) probability?

    Empirical probability is based on what did happen, calculated after events occur using observed data.

  • Define sample space in probability.

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

  • What is the complement of an event A?

    The complement of event A, written as A', is the set of all outcomes where event A does NOT occur.

  • What is the sum of probabilities of an event and its complement?

    The total probability is always 1, so \(P(A) + P(A') = 1\).

  • What are mutually exclusive events?

    Mutually exclusive events cannot happen at the same time; their intersection is empty.

  • How do you find the probability of mutually exclusive events A or B?

    For mutually exclusive events, \(P(A \cup B) = P(A) + P(B)\).

  • How do you find the probability of non-mutually exclusive events A or B?

    Use the addition rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\) to avoid double counting.

  • What defines independent events?

    Independent events are those where the occurrence of one does not affect the probability of the other.

  • How do you calculate the probability of independent events A and B occurring together?

    Multiply their probabilities: \(P(A \cap B) = P(A) \times P(B)\).

  • What is a contingency table used for?

    A contingency table displays frequencies across two categorical variables to find marginal, joint, and conditional probabilities.

  • Define conditional probability P(B|A).

    Conditional probability is the probability of event B occurring given that event A has occurred: \(P(B|A) = \frac{P(A \cap B)}{P(A)}\).

  • How do you calculate the probability of dependent events A and B occurring together?

    Multiply the probability of A by the conditional probability of B given A: \(P(A \cap B) = P(A) \times P(B|A)\).

  • What is Bayes' Theorem used for?

    Bayes' Theorem calculates the probability of an event based on prior knowledge of conditions related to the event.

  • State Bayes' Theorem formula.

    \(P(B|A) = \frac{P(A|B) \times P(B)}{P(A|B) \times P(B) + P(A|B') \times P(B')}\)

  • What is the Fundamental Counting Principle?

    When multiple events occur in sequence, multiply the number of options for each event to find total possible outcomes.

  • How do you calculate permutations of r objects from n distinct objects?

    \(P(n,r) = \frac{n!}{(n-r)!}\) counts ordered arrangements of r objects from n.

  • How do you calculate permutations when some objects are identical (non-distinct)?

    Divide total permutations by factorials of identical objects: \(\frac{n!}{r_1! r_2! \cdots}\).

  • What is the difference between permutations and combinations?

    Permutations consider order important; combinations do not.

  • How do you calculate combinations of r objects from n?

    \(C(n,r) = \frac{n!}{r!(n-r)!}\) counts unordered selections of r objects from n.