"In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.
20. Selecting an ace from a standard deck of 52 playing cards, and then selecting a jack from the deck without replacing the ace"

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Understand the concept of independence: Two events are independent if the occurrence of one event does not affect the probability of the other event occurring. Conversely, events are dependent if the outcome of one event influences the outcome of the other.

Identify the two events in the problem: Event A is selecting an ace from a standard deck of 52 cards, and Event B is selecting a jack from the deck without replacing the ace.

Analyze the situation: If the ace is not replaced after being selected, the total number of cards in the deck decreases from 52 to 51. This change affects the probability of selecting a jack in the second draw, as there are now fewer cards to choose from.

Determine dependency: Since the probability of selecting a jack (Event B) depends on whether the ace was removed (Event A), the two events are dependent.

Explain the reasoning: The dependency arises because the removal of the ace alters the composition of the deck, thereby influencing the likelihood of drawing a jack in the second draw.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Independent Events

Independent events are those whose outcomes do not affect each other. In probability, two events A and B are independent if the occurrence of A does not change the probability of B occurring. For example, flipping a coin and rolling a die are independent events because the result of one does not influence the other.

Dependent Events

Dependent events are those where the outcome of one event affects the outcome of another. In probability, two events A and B are dependent if the occurrence of A changes the probability of B occurring. For instance, drawing cards from a deck without replacement is a classic example of dependent events, as the first draw alters the composition of the deck for the second draw.

Probability without Replacement

When selecting items without replacement, the total number of items decreases with each selection, which affects the probabilities of subsequent selections. In the context of the question, after selecting an ace from a deck of cards, there are now only 51 cards left, which changes the probability of drawing a jack next. This illustrates how the first event influences the second, indicating dependence.

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