Statistics and Probability Quiz

Frequency Distribution and Relative Frequency

• A frequency distribution table details the grades in a Statistics class, indicating the number of occurrences for each mark.
• Example marks are recorded, ranging from 60 to 99 with their corresponding frequencies.
• The relative frequency of obtaining a mark of 90 is calculated as 5 divided by 63, resulting in approximately 0.079.
• To compute the relative frequency for marks less than or equal to 83, sum the frequencies from 60 to 83, yielding a relative frequency of approximately 0.31.

Classical Definition of Probability

• In an equally likely scenario for a sample space S, the probability of an event A is determined by the formula:
  P(A) = Number of outcomes in A / Total number of outcomes.
• This formula can also be expressed as:
  P(A) = n(A) / n(S), where n(A) represents the number of ways event A can occur, and n(S) represents the total outcomes in the sample space.

Example Scenario

• In a box containing 5 balls (3 red, 2 black), the probability of selecting a red ball is 3 out of 5, or 0.6.
• If rolling a die, the chance of not rolling a 6 can be calculated as 1 - (1/6) = 5/6.

Probability Properties

• The probability of an empty set (impossible event), denoted P(φ), equals 0.
• For events A and B, the union (addition rule) is represented as:
  P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
• If event A is a subset of B (A ⊆ B), then P(A) is less than or equal to P(B).

Theorems in Probability

• For mutually exclusive events A and B, the intersection probability P(A ∩ B) equals 0, simplifying the union to:
  P(A ∪ B) = P(A) + P(B).
• For events A, B, and C, the probability of the union is expressed as:
  P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C).

Worked Example of Probabilities

• Given P(A) = 1/2, P(B) = 3/1, P(C) = 1/4, with A ∩ C = φ, B ∩ C = φ, and P(A ∩ B) = 6/1:
  • Calculate P[(A ∩ B)'] as 1 - P(A ∩ B) = 5/6.
  • For P(A ∩ B'), use the relation where A = (A ∩ B') ∪ (A ∩ B) to find P(A ∩ B').
  • Determine P[(A ∪ B)'] and P(A' ∩ B') using similar calculations.

Axioms of Probability

• Axiom 1 states probabilities must fall between 0 and 1 (0 ≤ P(A) ≤ 1).
• Axiom 2 asserts P(S) = 1 and P(φ) = 0; the outcome will be among the simple outcomes.
• Axiom 3 states that for mutually exclusive events A and B:
  P(A ∪ B) = P(A) + P(B).

Infinite Sample Spaces

• For an infinite sequence of mutually exclusive events:
  P(A1 ∪ A2 ∪ A3 ∪ ...) = ΣP(Ai) from i=1 to ∞.

Elementary Properties of Probability

• Property 1 defines the probability of the complement of an event:
  P(Ā) = 1 - P(A), indicating the probability that event A does not occur.

Test your understanding of frequency distribution and the classical definition of probability. This quiz covers concepts like relative frequency calculation and the basic probability formula, helping you grasp essential statistical principles.