Cambridge: Probability and Statistics

This chapter introduces the mathematical methods for dealing with uncertainty and analysing data. You will learn to calculate the probability of events using fundamental rules and visual aids like Venn diagrams and tree diagrams. The main sub-topics include basic probability, the addition rule, and mutually exclusive and independent events. You will also work with statistical measures for a set of data, such as the mean, median, variance, and standard deviation.

The material becomes more abstract with the introduction of conditional probability, which deals with how the probability of an event changes based on the occurrence of another. This leads to the concept of a discrete random variable, where you will learn to define a probability distribution and calculate its expected value (mean) and variance. The Binomial distribution is a specific and important example of a discrete probability distribution that you will study in detail.

Success in this chapter depends on a strong grasp of logical reasoning and careful reading of problem statements. Foundational skills in fractions, percentages, and basic algebra are necessary. The most common difficulty is translating a word problem into a correct mathematical model, especially for conditional probability. This chapter builds the foundation for more advanced topics like the Normal distribution and hypothesis testing.

Formulas for Cambridge: Probability and Statistics

Addition Rule for Probability: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ For any two events A and B.
Probability of Complementary Event: $P(A') = 1 - P(A)$ A' is the event that A does not occur.
Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ The probability of A occurring, given that B has occurred.
Independent Events: $P(A \cap B) = P(A)P(B)$ This holds if and only if events A and B are independent.
Expected Value of a Discrete Random Variable: $E(X) = \sum x P(X=x)$ Also denoted by μ. It is the mean of the distribution.
Variance of a Discrete Random Variable: $Var(X) = E(X^2) - [E(X)]^2$ Where E(X^2) = Σ x² P(X=x).
Binomial Probability Formula: $P(X=r) = \binom{n}{r} p^r (1-p)^{n-r}$ For a random variable X ~ B(n, p).
Expected Value of a Binomial Distribution: $E(X) = np$ For a random variable X ~ B(n, p).
Variance of a Binomial Distribution: $Var(X) = np(1-p)$ For a random variable X ~ B(n, p).