Statistics and Probability

In the study of probability, various visual tools and formulae are employed to simplify and understand complex events. One such tool is the Venn diagram, which visually represents the relationship between different sets. Similarly, tree diagrams depict sequential events, while sample space diagrams and tables of outcomes provide a comprehensive view of all possible outcomes of a random experiment.

When dealing with combined events, the probability of either event A or event B occurring is given by \( P(A \cup B) \). This is calculated as the sum of the probabilities of each event minus the probability of both events occurring simultaneously. On the other hand, if two events are mutually exclusive, meaning they cannot occur at the same time, their intersection probability \( P(A \cap B) \) is zero.

Conditional probability, denoted as \( P(A|B) \), represents the probability of event A occurring given that event B has already occurred. Lastly, if two events are independent, the occurrence of one does not affect the occurrence of the other, and their intersection probability is the product of their individual probabilities.

Here are the key formulae:

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) \\ P(A \cap B) = 0 \quad \text{(For mutually exclusive events)} \\ P(A|B) = \frac{P(A \cap B)}{P(B)} \\ P(A \cap B) = P(A)P(B) \quad \text{(For independent events)} $$

Let's consider an example to illustrate these concepts:

Suppose we have two events: A, which represents drawing a red card from a standard deck of cards, and B, which represents drawing a face card (Jack, Queen, or King). The probability of drawing a red card, \( P(A) \), is \(\frac{26}{52}\) or \(\frac{1}{2}\). The probability of drawing a face card, \( P(B) \), is \(\frac{12}{52}\) or \(\frac{3}{13}\). The probability of drawing a red face card (intersection of A and B), \( P(A \cap B) \), is \(\frac{6}{52}\) or \(\frac{3}{26}\). Using the formula for combined events:

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) \\ P(A \cup B) = \frac{1}{2} + \frac{3}{13} - \frac{3}{26} = \frac{16}{26} \text{ or } \frac{8}{13} $$

Thus, the probability of drawing either a red card or a face card from a standard deck is \(\frac{8}{13}\).

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