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Introductory Statistics: Probability and Counting

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이 집합의 용어 (30)

  • What is a probability experiment?
    An action or trial where specific results are obtained.
  • Define an outcome in probability.
    The result of a single trial in a probability experiment.
  • What is a sample space?
    The set of all possible outcomes in a probability experiment.
  • What is an event in probability?
    One or more outcomes; a subset of the sample space.
  • State the fundamental counting principle.
    If one event can occur in m ways and a second in n ways, the total ways both can occur in sequence is \(m \times n\).
  • How many 4-digit access codes are possible if digits cannot be repeated?
    10 × 9 × 8 × 7 = 5040 possible codes.
  • How many 4-digit access codes are possible if digits can be repeated?
    10 × 10 × 10 × 10 = 10,000 possible codes.
  • How many 4-digit access codes if the first digit cannot be 0 or 1 but digits can repeat?
    8 × 10 × 10 × 10 = 8,000 possible codes.
  • Define classical (theoretical) probability.
    Probability where each outcome in the sample space is equally likely.
  • Formula for classical probability of event E.
    \(P(E) = \frac{\text{Number of outcomes in } E}{\text{Number of outcomes in sample space}}\)
  • What is empirical (statistical) probability?
    Probability based on observed data or relative frequency from experiments.
  • Formula for empirical probability of event E.
    \(P(E) = \frac{\text{Frequency of event } E}{\text{Total frequency}}\)
  • What is subjective probability?
    Probability based on intuition, educated guesses, or estimates.
  • What is the range of probabilities for any event E?
    The probability satisfies \(0 \leq P(E) \leq 1\).
  • Define the complement of an event E.
    All outcomes in the sample space not in event E, denoted E'.
  • Relationship between probabilities of an event and its complement.
    \(P(E) + P(E') = 1\)
  • What is conditional probability?
    The probability of event B occurring given event A has occurred, denoted \(P(B|A)\).
  • Formula for conditional probability of B given A.
    \(P(B|A) = \frac{P(A \text{ and } B)}{P(A)}\)
  • Define independent events.
    Events where the occurrence of one does not affect the probability of the other.
  • Multiplication rule for probability of A and B (dependent events).
    \(P(A \text{ and } B) = P(A) \times P(B|A)\)
  • Multiplication rule for independent events A and B.
    \(P(A \text{ and } B) = P(A) \times P(B)\)
  • What does it mean if two events are mutually exclusive?
    They cannot occur at the same time; no outcomes in common.
  • Addition rule for probability of A or B.
    \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)
  • Simplified addition rule for mutually exclusive events A and B.
    \(P(A \text{ or } B) = P(A) + P(B)\)
  • Define permutation.
    An ordered arrangement of objects.
  • Formula for number of permutations of n distinct objects.
    \(n! = n \times (n-1) \times (n-2) \times ... \times 1\)
  • What is 0! equal to?
    0! = 1 by definition.
  • Formula for permutations of n objects taken r at a time.
    \(nPr = \frac{n!}{(n-r)!}\)
  • Define combination.
    A selection of r objects from n without regard to order.
  • Formula for combinations of n objects taken r at a time.
    \(nCr = \frac{n!}{r! (n-r)!}\)