Conditional Probability and Independent Events

Study Notes

Definition: The probability of an event occurring given that another event has occurred.
Notation: P(A|B) reads as "the probability of A given B"
Formula: P(A|B) = P(A ∩ B) / P(B)
Properties:
- P(A|B) ≥ 0
- P(A|B) ≤ 1
- P(A|B) + P(A'|B) = 1
Chain Rule: P(A ∩ B ∩ C) = P(A) * P(B|A) * P(C|A ∩ B)

Definition: Two events are independent if the occurrence of one event does not affect the probability of the other event.
Notation: A and B are independent, denoted as A ⊥ B
Properties:
- P(A ∩ B) = P(A) * P(B)
- P(A|B) = P(A)
- P(B|A) = P(B)
Examples:
- Tossing a coin twice: the outcome of the first toss does not affect the outcome of the second toss.
- Drawing two cards from a deck: the first card drawn does not affect the probability of drawing a specific card on the second draw.

Definition: The process of making conclusions about a population based on a sample of data.
Types of Inference:
- Point Estimation: estimating a population parameter with a single value.
- Interval Estimation: estimating a population parameter with a range of values.
- Hypothesis Testing: testing a claim about a population parameter.
Key Concepts:
- Sample Space: the set of all possible outcomes of an experiment.
- Sampling Distribution: the distribution of a statistic over many samples.
- Confidence Interval: a range of values within which a population parameter is likely to lie.
- P-Value: the probability of obtaining a result as extreme or more extreme than the one observed, assuming that the null hypothesis is true.

Conditional probability calculates the probability of an event occurring given that another event has occurred.
Notated as P(A|B), read as "the probability of A given B".
Calculated using the formula: P(A|B) = P(A ∩ B) / P(B).
Properties:
- Always non-negative (P(A|B) ≥ 0).
- Always less than or equal to 1 (P(A|B) ≤ 1).
- Satisfies the law of total probability: P(A|B) + P(A'|B) = 1.
Chain Rule: P(A ∩ B ∩ C) = P(A) * P(B|A) * P(C|A ∩ B) for events A, B, and C.

Two events are independent if the occurrence of one event does not affect the probability of the other event.
Notated as A ⊥ B.
Properties:
- Joint probability is the product of individual probabilities: P(A ∩ B) = P(A) * P(B).
- Conditional probability is equal to individual probability: P(A|B) = P(A) and P(B|A) = P(B).
Examples:
- Tossing a coin twice: outcomes are independent.
- Drawing two cards from a deck: draws are independent.

The process of making conclusions about a population based on a sample of data.
Types of Inference:
- Point Estimation: estimating a population parameter with a single value.
- Interval Estimation: estimating a population parameter with a range of values.
- Hypothesis Testing: testing a claim about a population parameter.
Key Concepts:
- Sample Space: the set of all possible outcomes of an experiment.
- Sampling Distribution: the distribution of a statistic over many samples.
- Confidence Interval: a range of values within which a population parameter is likely to lie.
- P-Value: the probability of obtaining a result as extreme or more extreme than the one observed, assuming that the null hypothesis is true.