The provided lecture notes offer a structured and clear look into Sections 5.3 and 5.4 of probability theory. They perfectly balance mathematical formulas with practical examples, making complex concepts easier to digest for students or data enthusiasts.
Here is a breakdown of the key concepts covered in the presentation:
Independence vs. Dependence:
- Two events are considered independent if the occurrence of one event does not affect the probability of the other.
- Conversely, they are dependent if the first event's occurrence does affect the second event's probability.
- The notes also clarify a common misconception: disjoint events (events with no outcomes in common) are inherently not independent.
The Multiplication Rules:
- For independent events, the Multiplication Rule states that $P(E \text{ and } F) = P(E) \cdot P(F)$.
- When dealing with dependent events, the General Multiplication Rule applies: $P(E \text{ and } F) = P(E) \cdot P(F|E)$.
- This means you multiply the probability of event E by the probability of event F occurring given that E has already occurred.
Conditional Probability:
- Conditional probability is denoted as $P(F|E)$, which reads as "the probability of event F given event E".
- It is calculated by dividing the probability of both E and F occurring by the probability of E occurring: $P(F|E) = \frac{P(E \text{ and } F)}{P(E)}$.
Practical Applications & Shortcuts:
- The notes highlight how to solve for "at-least" probabilities, showing that the probability of at least one event occurring is equal to $1 - P(\text{none occur})$.
- They also offer a handy rule of thumb for sampling: if a sample size is less than 5% of the population size, it is reasonable to treat the events as independent, even if sampled without replacement.