Introduction to Probability (Sections 6.1 & 6.2)

Welcome to probability! In this lesson, we'll cover the basics of probability, including understanding random experiments, sample spaces, and events. We will also explore the addition rules for probability, which help us calculate the likelihood of compound events.

Key Concepts from Section 6.1: Basic Probability

  • Random Experiment: An activity or phenomenon with a single distinct outcome for each trial, an uncertain outcome, and a specifiable sample space.
  • Outcome: Any member of the sample space.
  • Event: A set of outcomes.
  • Sample Space (S): The set of all possible distinct outcomes of an experiment. For example, tossing a coin has a sample space $S = \{H, T\}$. Tossing a coin three times and observing the *number* of heads yields a sample space $S = \{0, 1, 2, 3\}$.
  • Relative Frequency: If an experiment is performed $n$ times and event $A$ happens $k$ times, the relative frequency of $A$ is given by: $$ \text{Relative Frequency of A} = \frac{k}{n} $$
  • As $n$ increases, the relative frequency converges to the probability of $A$.
  • Classical Probability: The probability of an event $A$, denoted $P(A)$, is calculated as: $$P(A) = \frac{\text{number of outcomes in A}}{\text{total number of outcomes in the sample space}}$$.

Example: Imagine tossing a coin three times. The sample space, showing all 8 equally likely outcomes, is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

Probability Laws

  1. A probability of zero means the event cannot happen.
  2. A probability of one means the event must happen.
  3. All probabilities must be between zero and one, inclusive: $0 \le P(A) \le 1$.
  4. The sum of the probabilities of all outcomes in the sample space must equal one. If $P(A_i)$ is the probability of event $A_i$, and there are $n$ such outcomes, then $$P(A_1) + P(A_2) + ... + P(A_n) = 1$$.

Key Concepts from Section 6.2: Addition Rules for Probability

  • Compound Event: An event defined by combining two or more events.
  • Union (A ∪ B): The set of outcomes included in event $A$ or event $B$ or both. It is read "A union B".
  • Intersection (A ∩ B): The set of all outcomes included in both $A$ and $B$. It is read "A intersect B".
  • Complement (Ac): The set of all outcomes in the sample space that are *not* in $A$.
  • Complement of an Event: The probability of $A^c$ is given by $P(A^c) = 1 - P(A)$.

The General Addition Rule

For any two events $A$ and $B$, the probability of their union is given by:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Odds

  • The odds in favor of an event A occurring is given by $\frac{P(A)}{P(\text{not }A)} = \frac{P(A)}{P(A^c)}$.
  • The odds against an event A occurring is given by $\frac{P(\text{not }A)}{P(A)} = \frac{P(A^c)}{P(A)}$.

Keep practicing these concepts, and you'll become a probability pro in no time! Good luck, and remember to ask questions if you get stuck.