"1. What is the difference between independent and dependent events?

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Independent events are events where the occurrence of one event does not affect the probability of the other event occurring. For example, flipping a coin and rolling a die are independent events because the outcome of the coin flip does not influence the outcome of the die roll.

Dependent events are events where the occurrence of one event affects the probability of the other event occurring. For example, drawing two cards from a deck without replacement is a dependent event because the outcome of the first draw changes the probabilities for the second draw.

To mathematically express independence, two events A and B are independent if and only if P(A ∩ B) = P(A) × P(B), where P(A ∩ B) is the probability of both events occurring, and P(A) and P(B) are the probabilities of each event individually.

For dependent events, the probability of one event occurring given that another event has occurred is expressed using conditional probability: P(A | B) = P(A ∩ B) / P(B), where P(A | B) is the probability of A given B, P(A ∩ B) is the joint probability, and P(B) is the probability of B.

Understanding the difference between independent and dependent events is crucial for solving problems involving probability, as it determines whether you can multiply probabilities directly or need to account for conditional relationships.

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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 likelihood of B occurring, and vice versa. For example, flipping a coin and rolling a die are independent events; the result of one does not influence the other.

Dependent Events

Dependent events are events 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 likelihood of B occurring. For instance, drawing cards from a deck without replacement creates dependent events, as the first draw alters the composition of the deck for the second draw.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. It quantifies uncertainty and is foundational in statistics. For independent events, the probability of both events occurring is the product of their individual probabilities, while for dependent events, the probability must account for the influence of the first event on the second.