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Imagine that you're trying to understand the likelihood of an event, knowing that another event has already occurred. This is where conditional probability comes in. **Conditional probability** deals with the probability of event A happening given that event B has occurred. It's noted as \( P(A | B) \). A key formula for calculating this is:

• \( P(A | B) = \frac{P(A \cap B)}{P(B)} \)

where \( P(A \cap B) \) is the probability that both events A and B occur, and \( P(B) \) is the probability of event B occurring. Remember, this formula only works if \( P(B) > 0 \). Conditional probability is useful in fields like insurance and finance, where predictions based on existing knowledge are necessary.
Understanding how probability adjusts due to existing conditions helps in decision-making and assessing risk. It's essential to not confuse it with simply multiplying the individual probabilities of two independent events.

When discussing probability, you often hear about complementary events. But what exactly are they? Complementary events are two outcomes that cover all possibilities of a given situation. If one happens, the other cannot, and vice versa. For example, if event B is passing an exam, then the complementary event, \( \operatorname{not} B \), would be failing the exam.

• Complementary events have probabilities that add up to 1: \( P(B) + P(\operatorname{not} B) = 1 \).

This makes understanding these events straightforward because once you know one probability, you can easily find its complement by subtracting it from 1.
Whenever you're tasked with finding the probability of a situation not happening, remember this rule: compute the probability of the original event, and subtract it from 1. It’s a handy rule that simplifies calculations, especially in cases with many outcomes.

Probability is governed by fundamental rules that help in the calculation of complex probabilities. These **rules of probability** form the basis of calculating probabilities accurately.One of the primary rules is the **addition rule for non-mutually exclusive events**. It is used to determine the probability of either event A or event B occurring:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

Here, \( P(A \cup B) \) is the probability that either event A or B (or both) occurs. The term \( P(A \cap B) \) is subtracted to avoid double accounting in non-mutually exclusive events where both can happen.
Another crucial rule is the **multiplication rule for independent events** which is used when two events do not affect each other's outcome:

• \( P(A \cap B) = P(A) \times P(B) \)

These rules provide a framework for calculating probabilities efficiently and effectively. Understanding them helps interpret real-world situations and make informed predictions.