Questions and Answers
What type of probability is Sunil assessing when he believes he has a 50% chance of obtaining tails?
What type of probability does John use when he believes he has a 90% chance of receiving straight A's?
What type of probability is represented by the announcement of a 40% chance of a person being a Republican in the conference room?
Find P(D) when a sample space S yields five equally likely events, A, B, C, D, and E.
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What is P(B^c) when B^c consists of the events A, C, D, and E?
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Calculate P(A U C U E) given that P(A) = P(C) = P(E) = 1/5.
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The probability that Jane Peterson tells her friends the Amtrak train will arrive on time is _____ based on her past experiences.
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Why would Jane's estimated probability of the train arriving on time be considered inaccurate?
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Calculate P(A ∩ B) given P(A) = 0.57, P(B) = 0.22, and P(A | B) = 0.37.
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Calculate P(A U B) given P(A) = 0.57, P(B) = 0.22, and P(A ∩ B) = 0.081.
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Calculate P(B | A) given P(A ∩ B) = 0.081 and P(A) = 0.57.
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Are events A and B independent if P(A) = 0.50, P(B) = 0.45, and P(A ∩ B) = 0.31?
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Are events A and B mutually exclusive if P(A ∩ B) = 0?
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What is P((A U B)c) given P(A) = 0.50, P(B) = 0.45, and P(A ∩ B) = 0.31?
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What is the probability that at least one of the stocks will rise in price given P(A) = 0.61 and P(B) = 0.39?
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Are events A and B mutually exclusive if P(A ∩ B) ≠ 0?
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Are events A and B independent if P(A | B) = P(A)?
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What is the probability that neither the seven-year AA-rated bond nor the seven-year A-rated bond defaults?
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What is the probability that at least one of the bonds defaults given the individual probabilities?
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Study Notes
Probability Concepts
- Classical Probability: Determined when outcomes are equally likely, e.g., flipping a fair coin results in a 50% chance of tails.
- Subjective Probability: Based on personal belief or assessment, such as estimating a 90% chance of achieving straight A's.
- Empirical Probability: Derived from observed data or experimental outcomes, e.g., reporting a 50% chance of a train being on time based on past experiences.
Probability Calculations
- Probability of a Single Event (P(D)): For five equally likely events, the probability of any single event (e.g., P(D)) is 0.20.
- Complementary Events: The probability of the complement of an event (e.g., P(B^c)) is calculated by subtracting the probability of the event from 1, resulting in 0.80 for P(B^c).
- Union of Events (P(A U C U E)): The probability of the union of several events can be calculated by summing their individual probabilities; e.g., P(A) + P(C) + P(E) results in 0.60.
Advanced Probability
- Calculating Joint Probability (P(A ∩ B)): The joint probability of two events can be derived using conditional probability; for example, P(A ∩ B) = P(A | B) * P(B) resulting in 0.081.
- Calculating Union (P(A U B)): The combined probability of two events is calculated using P(A) + P(B) − P(A ∩ B), yielding 0.709.
- Conditional Probability (P(B | A)): Conditional probability shows the likelihood of an event given another event has occurred, calculated as P(A ∩ B) / P(A), equating to 0.142.
Independence and Mutual Exclusivity
- Independent Events: Events A and B are independent if P(A | B) equals P(A); in this case, they are not independent.
- Mutually Exclusive Events: Events that cannot happen simultaneously have a joint probability of zero; however, A and B are not mutually exclusive since P(A ∩ B) is not zero.
Probability of Non-occurrence
- Calculating neither A nor B Occurring: To calculate the probability that neither event occurs, use the formula P((A U B)^c) = 1 − P(A U B), resulting in 0.36.
Stock Price Probabilities
- At least one Event Occurring: The probability of at least one stock rising in price is found by calculating P(A U B) = P(A) + P(B) − P(A ∩ B), yielding 0.77.
Bond Default Probabilities
- Probability of Neither Bond Defaulting: To find the likelihood that neither bond defaults, use P((A U B)^c) = 1 − P(A U B), resulting in 0.10.
- Probability of at least One Bond Defaulting: The combined probability of either bond defaulting can be calculated similarly, yielding a probability of 0.90.
Summary of Probability Types
- Events classified as empirical, classical, or subjective variations affects their interpretation and usage in practical applications.
- Understanding complements, joint probabilities, and conditional relationships forms the foundation for more complex probability scenarios.
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Description
Test your understanding of key concepts in Chapter 4 of ISDS. This quiz covers essential topics such as classical and subjective probability, with real-world examples for application. Perfect for reinforcing your knowledge before exams!
