for which of the following probability assignments are events a and b independent

 To determine if events A and B are independent in probability theory, we use the definition of independence: Events A and B are independent if the occurrence of one does not affect the probability of the other.

If P(A) denotes the probability of event A occurring, and P(B) denotes the probability of event B occurring, and P(A ∩ B) represents the probability of both A and B occurring:

  1. Rolling a Fair Die: Events like rolling a fair die and getting a certain number are typically independent. For example, getting a 4 on one roll doesn't affect the probability of getting a 6 on the next roll. So, events A and B (getting a specific number on two rolls) are usually independent.

  2. Flipping a Coin: Similarly, flipping a fair coin has independent events. Getting heads on one flip doesn't affect the probability of getting heads on the next flip. Thus, events A and B (getting heads on two flips) are typically independent.

  3. Drawing Cards from a Deck: If you draw a card from a standard deck and replace it before drawing the second card, the events are usually independent. For instance, drawing a heart on the first draw doesn't affect the probability of drawing a spade on the second draw.

However, if the card is not replaced after drawing (without replacement), the events become dependent. For instance, drawing a heart on the first draw affects the number of hearts left in the deck for the second draw, altering the probability.

Therefore, for assignments involving a fair die roll or a fair coin flip, and when events involve independent trials (like drawing with replacement), events A and B are likely to be independent. For drawing cards without replacement, events may not be independent.

Remember, the concept of independence in probability relies on whether the occurrence of one event affects the probability of the other

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