Stats 1 Week 8

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Probability Notes

Types of Probability

Marginal Probability

Definition: The probability of a single event occurring without consideration of any other events.

Formula: P(A)

Example: The probability of drawing a red card from a deck of cards, P(Red) = 26/52 = 0.5.

Joint Probability

Definition: The probability of two or more events occurring simultaneously.

Formula: P(A ∩ B)

Example: The probability of drawing a red card that is also a four, P(Red ∩ Four) = 2/52 = 1/26.

Conditional Probability

Definition: The probability of an event occurring given that another event has already occurred.

Formula: P(A|B) = P(A ∩ B) / P(B)

Example: The probability of drawing a four given that a red card has been drawn, P(Four|Red) = P(Four ∩ Red) / P(Red) = (2/52) / (26/52) = 1/13.

Multiplicity Rules

Addition Rule

For any two events A and B:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

If A and B are mutually exclusive (disjoint):

P(A ∪ B) = P(A) + P(B)

Multiplication Rule

For any two events A and B:

P(A ∩ B) = P(A) * P(B|A)

If A and B are independent:

P(A ∩ B) = P(A) * P(B)

Independent vs. Disjoint Events

Independent Events

Two events A and B are independent if the occurrence of one does not affect the occurrence of the other.

Formula: P(A ∩ B) = P(A) * P(B)

Example: Flipping a coin and rolling a die.

Disjoint (Mutually Exclusive) Events

Two events A and B are disjoint if they cannot occur at the same time.

Formula: P(A ∩ B) = 0

Example: Drawing a card that is both a heart and a club.

Law of Total Probability

Definition: A way to find the probability of an event based on the probabilities of related events.

Formula: P(A) = Σ P(A|B_i) * P(B_i)

Explanation: If B₁, B₂, ..., B_n are mutually exclusive and exhaustive events, then the probability of A can be found by summing the probabilities of A given each B_i multiplied by the probability of B_i.

Bayes' Theorem

Definition: A way to update the probability of an event based on new evidence.

Formula: P(A|B) = P(B|A) * P(A) / P(B)

Explanation: Bayes' theorem allows us to reverse conditional probabilities. For example, if we know P(B|A), P(A), and P(B), we can find P(A|B).

Summary of Formulas

• Marginal Probability: P(A)
• Joint Probability: P(A ∩ B)
• Conditional Probability: P(A|B) = P(A ∩ B) / P(B)
• Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Multiplication Rule: P(A ∩ B) = P(A) * P(B|A)
• Law of Total Probability: P(A) = Σ P(A|B_i) * P(B_i)
• Bayes' Theorem: P(A|B) = P(B|A) * P(A) / P(B)