Flashcards on Pairwise Independent and Mutually Independent

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Define pairwise independence in the context of probability theory.

A collection of events is said to be pairwise independent if every pair of events is independent, meaning the occurrence or non-occurrence of one event does not affect the probability of the other event.

What is the definition of mutual independence in probability?

A collection of events is said to be mutually independent if every possible subset of the events is independent, meaning the occurrence or non-occurrence of any combination of events does not affect the probability of the others.

What is the main difference between pairwise independence and mutual independence?

Pairwise independence only considers the independence of pairs of events, while mutual independence extends this concept to all possible combinations of events.

Give an example of a set of events that are pairwise independent but not mutually independent.

Consider three events: A, B, and C. If P(A) = P(B) = P(C) = 1/2, and P(A ∩ B) = P(B ∩ C) = P(C ∩ A) = 1/4, then A, B, and C are pairwise independent but not mutually independent.

Can a set of mutually independent events also be pairwise independent?

Yes, if a set of events is mutually independent, then it is also pairwise independent. However, the converse is not necessarily true.

What is the relationship between pairwise independent events and mutually independent events?

Every collection of mutually independent events is pairwise independent, but not every collection of pairwise independent events is mutually independent.

True or False: If three events are pairwise independent, they must be mutually independent.

False. The pairwise independence of events does not imply their mutual independence.

In a deck of playing cards, are drawing two cards from the deck considered pairwise independent events?

Yes, drawing two cards from a well-shuffled deck can be considered pairwise independent events since the outcome of drawing one card does not affect the probability of the other card being drawn.

Give an example of a set of events that are mutually independent but not pairwise independent.

Consider three events: A, B, and C. If P(A) = P(B) = P(C) = 1/4 and P(A ∩ B ∩ C) = 1/8, then A, B, and C are mutually independent but not pairwise independent.

Can two events be mutually independent if they are not pairwise independent?

No, if two events are not pairwise independent, they cannot be mutually independent. Pairwise independence is a necessary condition for mutual independence.

What is the key concept in probability theory that relates to pairwise independence and mutual independence?

Independence. Both pairwise independence and mutual independence are based on the notion of events being independent of each other.

How can you determine whether a set of events is pairwise independent?

To determine pairwise independence, you need to check whether the probability of the intersection of any two events equals the product of their individual probabilities.

What is an example of a practical application of pairwise independent events?

In cryptography, the randomness of pairwise independent events is utilized to ensure secure encryption algorithms and key generation.

Are pairwise independent events always mutually exclusive?

No, pairwise independent events can overlap and have non-zero intersection probabilities. Mutually exclusive events are a separate concept.

What is the probability of three pairwise independent events A, B, and C occurring together?

If A, B, and C are pairwise independent, the probability of their intersection P(A ∩ B ∩ C) is equal to the product of their individual probabilities: P(A) × P(B) × P(C).

Define pairwise independence in the context of probability theory.

A collection of events is said to be pairwise independent if every pair of events is independent, meaning the occurrence or non-occurrence of one event does not affect the probability of the other event.

What is the definition of mutual independence in probability?

A collection of events is said to be mutually independent if every possible subset of the events is independent, meaning the occurrence or non-occurrence of any combination of events does not affect the probability of the others.

What is the main difference between pairwise independence and mutual independence?

Pairwise independence only considers the independence of pairs of events, while mutual independence extends this concept to all possible combinations of events.

Give an example of a set of events that are pairwise independent but not mutually independent.

Consider three events: A, B, and C. If P(A) = P(B) = P(C) = 1/2, and P(A ∩ B) = P(B ∩ C) = P(C ∩ A) = 1/4, then A, B, and C are pairwise independent but not mutually independent.

Can a set of mutually independent events also be pairwise independent?

Yes, if a set of events is mutually independent, then it is also pairwise independent. However, the converse is not necessarily true.

What is the relationship between pairwise independent events and mutually independent events?

Every collection of mutually independent events is pairwise independent, but not every collection of pairwise independent events is mutually independent.

True or False: If three events are pairwise independent, they must be mutually independent.

False. The pairwise independence of events does not imply their mutual independence.

In a deck of playing cards, are drawing two cards from the deck considered pairwise independent events?

Yes, drawing two cards from a well-shuffled deck can be considered pairwise independent events since the outcome of drawing one card does not affect the probability of the other card being drawn.

Give an example of a set of events that are mutually independent but not pairwise independent.

Consider three events: A, B, and C. If P(A) = P(B) = P(C) = 1/4 and P(A ∩ B ∩ C) = 1/8, then A, B, and C are mutually independent but not pairwise independent.

Can two events be mutually independent if they are not pairwise independent?

No, if two events are not pairwise independent, they cannot be mutually independent. Pairwise independence is a necessary condition for mutual independence.

What is the key concept in probability theory that relates to pairwise independence and mutual independence?

Independence. Both pairwise independence and mutual independence are based on the notion of events being independent of each other.

How can you determine whether a set of events is pairwise independent?

To determine pairwise independence, you need to check whether the probability of the intersection of any two events equals the product of their individual probabilities.

What is an example of a practical application of pairwise independent events?

In cryptography, the randomness of pairwise independent events is utilized to ensure secure encryption algorithms and key generation.

Are pairwise independent events always mutually exclusive?

No, pairwise independent events can overlap and have non-zero intersection probabilities. Mutually exclusive events are a separate concept.

What is the probability of three pairwise independent events A, B, and C occurring together?

If A, B, and C are pairwise independent, the probability of their intersection P(A ∩ B ∩ C) is equal to the product of their individual probabilities: P(A) × P(B) × P(C).