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What is the formula for conditional probability?
What is the formula for conditional probability?
What is the property of conditional probability that states P(A|B) + P(A'|B) = 1?
What is the property of conditional probability that states P(A|B) + P(A'|B) = 1?
Which of the following events are independent?
Which of the following events are independent?
What is the purpose of statistical inference?
What is the purpose of statistical inference?
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What is the type of inference that involves testing a claim about a population parameter?
What is the type of inference that involves testing a claim about a population parameter?
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What is the term for the distribution of a statistic over many samples?
What is the term for the distribution of a statistic over many samples?
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What is the term for a range of values within which a population parameter is likely to lie?
What is the term for a range of values within which a population parameter is likely to lie?
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What is the probability of obtaining a result as extreme or more extreme than the one observed, assuming that the null hypothesis is true?
What is the probability of obtaining a result as extreme or more extreme than the one observed, assuming that the null hypothesis is true?
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What is the chain rule of probability?
What is the chain rule of probability?
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What is the notation for two independent events?
What is the notation for two independent events?
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Study Notes
Conditional Probability
- 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)
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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)
Independent Events
- 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
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Properties:
- P(A ∩ B) = P(A) * P(B)
- P(A|B) = P(A)
- P(B|A) = P(B)
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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.
Statistical Inference
- Definition: The process of making conclusions about a population based on a sample of data.
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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.
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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
- 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.
Independent Events
- 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.
Statistical Inference
- 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.
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Description
Quiz on conditional probability, its definition, notation, formula, and properties, as well as independent events and the chain rule.