Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
11. Returning a rented movie after the due date and receiving a late fee

Verified step by step guidance

1. Understand the definition of independent and dependent events: Independent events are those where the occurrence of one event does not affect the probability of the other. Dependent events are those where the occurrence of one event affects the probability of the other.
2. Identify the two events in the problem: Event A is 'returning a rented movie after the due date,' and Event B is 'receiving a late fee.'
3. Analyze the relationship between the two events: Consider whether Event A (returning the movie late) influences the likelihood of Event B (receiving a late fee).
4. Determine if the events are dependent: If returning the movie late directly causes or increases the likelihood of receiving a late fee, then the events are dependent. If there is no such relationship, they would be independent.
5. Conclude and explain: Based on the analysis, classify the events as dependent or independent and provide reasoning. For example, if returning the movie late directly results in a late fee, the events are dependent because one event affects the outcome of the other.

Key Concepts

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 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 likelihood of B occurring. For instance, drawing cards from a deck without replacement is a classic example of dependent events, as the first draw affects the composition of the deck for the second draw.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which represents the probability of event A occurring given that event B has occurred. Understanding conditional probability is crucial for determining whether events are independent or dependent, as it helps to analyze how the occurrence of one event influences the likelihood of another.