ISDS Chapter 4 Flashcards

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Questions and Answers

What type of probability is Sunil assessing when he believes he has a 50% chance of obtaining tails?

Classical probability (correct)

Subjective probability

Empirical probability

None of the above

What type of probability does John use when he believes he has a 90% chance of receiving straight A's?

Classical probability

Empirical probability

Theoretical probability

Subjective probability (correct)

What type of probability is represented by the announcement of a 40% chance of a person being a Republican in the conference room?

Classical probability

Empirical probability (correct)

Theoretical probability

Subjective probability

Find P(D) when a sample space S yields five equally likely events, A, B, C, D, and E.

0.20 Signup and view all the answers

What is P(B^c) when B^c consists of the events A, C, D, and E?

0.80 Signup and view all the answers

Calculate P(A U C U E) given that P(A) = P(C) = P(E) = 1/5.

0.60 Signup and view all the answers

The probability that Jane Peterson tells her friends the Amtrak train will arrive on time is _____ based on her past experiences.

0.50 Signup and view all the answers

Why would Jane's estimated probability of the train arriving on time be considered inaccurate?

The experiment must be repeated a large number of times for empirical probabilities to be accurate. Signup and view all the answers

Calculate P(A ∩ B) given P(A) = 0.57, P(B) = 0.22, and P(A | B) = 0.37.

0.081 Signup and view all the answers

Calculate P(A U B) given P(A) = 0.57, P(B) = 0.22, and P(A ∩ B) = 0.081.

0.709 Signup and view all the answers

Calculate P(B | A) given P(A ∩ B) = 0.081 and P(A) = 0.57.

0.142 Signup and view all the answers

Are events A and B independent if P(A) = 0.50, P(B) = 0.45, and P(A ∩ B) = 0.31?

False Signup and view all the answers

Are events A and B mutually exclusive if P(A ∩ B) = 0?

True Signup and view all the answers

What is P((A U B)c) given P(A) = 0.50, P(B) = 0.45, and P(A ∩ B) = 0.31?

0.36 Signup and view all the answers

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?

0.77 Signup and view all the answers

Are events A and B mutually exclusive if P(A ∩ B) ≠ 0?

False Signup and view all the answers

Are events A and B independent if P(A | B) = P(A)?

True Signup and view all the answers

What is the probability that neither the seven-year AA-rated bond nor the seven-year A-rated bond defaults?

0.10 Signup and view all the answers

What is the probability that at least one of the bonds defaults given the individual probabilities?

0.90 Signup and view all the answers

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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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!