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In probability, when we say two events are independent, we mean that the occurrence of one event does not affect the occurrence of the other. For example, consider rolling a die and flipping a coin. These two events are independent because the result of the die roll does not influence the outcome of the coin flip, and vice versa.
In mathematical terms, independence between two events, say event \( E \) and event \( F \), is expressed using the equation:

• \( P(E \, \text{and} \, F) = P(E) \cdot P(F) \)

This means that the joint probability of both events occurring is simply the product of their individual probabilities. If changes in the occurrence probability for one event have no impact on the probability of the other, they are independent.
The exercise we are considering shows us another important condition for independence: if \( P(E \mid F) = P(E) \), meaning the probability of event \( E \) given that \( F \) has occurred is the same as \( E \) occurring without any condition, then \( P(F \mid E) = P(F) \). This confirms the bidirectional nature of independence because the events do not influence one another.

Event probability is a fundamental concept in probability theory that deals with the likelihood of an event happening. An event, in simple terms, is any outcome or set of outcomes of a particular process. For instance, drawing a card from a deck or tossing a coin can each be considered an event.
The probability of an event is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. For example:

• The probability of drawing a red card from a standard deck of cards is \( \frac{26}{52} = 0.5 \), since half of the 52 cards are red.
• The probability of rolling a 3 on a six-sided die is \( \frac{1}{6} \).

When working with probabilities, it's important to remember that the total probability of all possible outcomes should always sum to 1. This is often referred to as the sample space of possible events. In the context of the exercise, understanding event probability allows us to express conditional statements and relationships between events, making it crucial for deeper probability exploration.

Joint probability refers to the probability of two events occurring simultaneously. It's a way of combining probabilities of individual events and is more complex than dealing with single events alone.

• The joint probability of events \( E \) and \( F \) happening together is denoted as \( P(E \cap F) \).

This calculation is important because it gives insight into how two events interact with each other. In many practical situations, knowing the likelihood of events co-occurring can inform decisions and predictions.
In our exercise, the calculation of joint probability plays a central role. We see it in action when determining whether two events are independent. Recall from the provided solution steps that:

• When \( P(E \mid F) = P(E) \) and subsequently \( P(E \cap F) = P(E) \cdot P(F) \), this setup confirms the independence of \( E \) and \( F \).

Joint probability thus forms a crucial part of evaluating conditional probabilities and the overall structure of complex probabilistic scenarios.