3.2.4 Probability Formulae

Probability is a fundamental concept in statistics, and understanding the key formulae is essential for solving problems related to random events. Here's a summary of the most important probability formulae, along with explanations and examples.

Basic Probability Formula

The probability of an event $A$ is given by:

$$P(A) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$$

Complementary Probability

The probability of the complement of an event $A$ (i.e., the event not happening) is:

$$P(A') = 1 - P(A)$$

Addition Rule (Union of Two Events)

For any two events $A$ and $B$ , the probability that $A$ or $B$ (or both) occur is:

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

Multiplication Rule (Intersection of Two Events)

For independent events $A$ and $B$ , the probability that both $A$ and $B$ occur is:

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

For dependent events $A$ and $B$ , the probability that both $A$ and $B$ occur is:

$$P(A \cap B) = P(A) \times P(B \mid A)$$

Where $P(B \mid A)$ is the conditional probability of $B$ given that $A$ has occurred.

Conditional Probability

The probability of event $A$ occurring given that event $B$ has already occurred is:

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

Total Probability

If events $B_1, B_2, \ldots, B_n$ are mutually exclusive and exhaustive and $A$ is an event that can occur if any one of $B_1, B_2, \ldots, B_n$ occurs, then:

$$P(A) = P(A \cap B_1) + P(A \cap B_2) + \ldots + P(A \cap B_n)$$

Or, using the conditional probability:

$$P(A) = P(B_1) \times P(A \mid B_1) + P(B_2) \times P(A \mid B_2) + \ldots + P(B_n) \times P(A \mid B_n)$$

Bayes' Theorem

Bayes' Theorem allows the calculation of the probability of an event based on prior knowledge of conditions that might be related to the event. It is given by:

$$P(B_i \mid A) = \frac{P(B_i) \times P(A \mid B_i)}{P(A)}$$

Where $B_i$ is one of the mutually exclusive and exhaustive events, and $A$ is the event for which we want to reverse the conditional probability.

Probability of "At Least One"

The probability that at least one event occurs is given by:

$$P(\text{At least one}) = 1 - P(\text{None})$$

Summary

These probability formulae are essential tools for calculating the likelihood of various outcomes in different scenarios. Understanding when and how to apply each formula allows for accurate analysis and decision-making in both theoretical and practical situations. The examples provided illustrate how these formulae are used in real-world contexts, helping to clarify their application.

---