AP Statistic

  1. Probability

Basic probability concepts

  • Sample space (S): The set of all possible outcomes of a random experiment. For example, when flipping a coin, S = {Heads, Tails}.

  • Events: A subset of the sample space. For example, getting a head in a coin flip corresponds to the event {Heads}.

  • Probability of an event (P(E)): Calculated as the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. Mathematically, ( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} ).

Conditional probability and independence

  • Conditional probability (P(A|B)): The probability of event A occurring given that event B has occurred. It can be calculated using: ( P(A|B) = \frac{P(A \cap B)}{P(B)} ) if P(B) > 0.

  • Independence: Two events A and B are independent if the occurrence of one does not affect the occurrence of the other. Mathematically, this is expressed as ( P(A \cap B) = P(A) \cdot P(B) ).

Law of Large Numbers

  • Law of Large Numbers: States that as the number of trials increases, the empirical probability (relative frequency) of an event will converge to the theoretical probability.

Central Limit Theorem

  • Central Limit Theorem (CLT): In the context of inferential statistics, CLT states that regardless of the original distribution of the data, the distribution of the sample means will approximate a normal distribution as the sample size becomes large, typically n ≥ 30. This is crucial for conducting hypothesis tests and constructing confidence intervals for population means