Probability events can be independent or mutually exclusive, but rarely both. Independent events don't affect each other's likelihood, while mutually exclusive events can't happen together. Understanding these concepts is crucial for calculating probabilities accurately.

Sampling with and without replacement impacts event independence. With replacement keeps probabilities constant, while without replacement changes them. These ideas form the foundation for more advanced probability concepts like the Law of Total Probability and Bayes' Theorem.

Independent and Mutually Exclusive Events

Independence vs mutual exclusivity

• Independent events occur when the outcome of one event does not influence the probability of another event happening (rolling a 6 on a die, then flipping a coin and getting heads)
  • Probability of both events occurring together is the product of their individual probabilities: $P(A \cap B) = P(A) \times P(B)$
  • Conditional probability of one event given the other remains unchanged: $P(A|B) = P(A)$ and $P(B|A) = P(B)$
• Mutually exclusive events cannot happen at the same time (drawing a heart or a spade from a deck of cards in a single draw)
  • Probability of both events occurring simultaneously is always zero: $P(A \cap B) = 0$
  • Probability of either event occurring is the sum of their individual probabilities: $P(A \cup B) = P(A) + P(B)$
• Events can only be both independent and mutually exclusive if at least one event has a probability of zero (rolling a 7 on a standard six-sided die, then flipping a coin and getting tails)
• Venn diagrams can be used to visually represent the relationship between independent and mutually exclusive events

Sampling with and without replacement

• Sampling with replacement involves putting each selected item back before the next selection, keeping event probabilities constant (picking a marble from a bag, recording its color, then putting it back and repeating)
  • Each selection is an independent event
  • Probability of an event remains the same for each pick
• Sampling without replacement involves keeping selected items out for subsequent selections, changing event probabilities (dealing cards from a shuffled deck)
  • Probability of an event changes after each selection
  • Use the hypergeometric probability formula to calculate probabilities:
    • $P(X = k) = \frac{{K \choose k} {N-K \choose n-k}}{{N \choose n}}$
    • $N$ = total items in the population (52 cards in a standard deck)
    • $K$ = items with desired characteristic in the population (13 hearts in a standard deck)
    • $n$ = sample size (5 card poker hand)
    • $k$ = items with desired characteristic in the sample (2 hearts in a 5 card hand)

Classification of probability events

• Determine event independence by checking if:
  1. $P(A|B) = P(A)$ and $P(B|A) = P(B)$
  2. $P(A \cap B) = P(A) \times P(B)$
  • If both conditions hold true, the events are independent (drawing a king, then drawing a heart from a standard deck with replacement)
• Determine event mutual exclusivity by checking if:
  1. The events cannot occur at the same time
  2. $P(A \cap B) = 0$
  • If both conditions hold true, the events are mutually exclusive (drawing a king or drawing a queen from a standard deck in a single draw)
• Verify event independence and mutual exclusivity based on given probability values and event descriptions to accurately classify the relationship between events
• Independence tests can be used to statistically verify the independence of events in more complex scenarios

Advanced Probability Concepts

• Law of Total Probability: Used to calculate the probability of an event by considering all possible scenarios
• Bayes' Theorem: Allows for updating probabilities based on new information, connecting prior and posterior probabilities
• These concepts build upon the understanding of independent and mutually exclusive events to solve more complex probability problems