Vijay Maths Tutor (PGCE BSc PhD)

The binomial distribution  is a special type of discrete probability distribution . It is used to model situations where there are a fixed number of trials, each with two possible outcomes , and the probability of success is constant. Coin Toss Example Consider tossing a biased coin 4 times , where: P(head) = 0.3, P(tail) = 0.7 Let X be the number of heads obtained, so X is a discrete random variable . We are interested in its probability distribution: P(X = x) for x = 0

Introduction The Fibonacci sequence is defined using the recurrence formula: f n+2 = f n+1 + f n for n ≧ 1 with f 1 = 1 and f 2 = 1. The sequence begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ... In this post, we explore four elegant results that can be proven using mathematical induction  (including strong induction). 1. Every Fourth Fibonacci Number Is Divisible by 3 Theorem The subsequence f 4n is divisible by 3 for n ≧ 1. Proof (ordinary induction) When

Under certain conditions, a binomial distribution  (which is discrete ) can be well-approximated by a normal distribution  (which is continuous ). This approximation works well when: n is large, and p is not too close to 0 or 1 (ideally p ≈ 0.5 ). More precisely, we often use the rule of thumb: np ≥ 5 and n(1−p) ≥ 5 . Using the Approximation For n large and p close to 0.5, we can approximate as follows So the mean and variance, μ = np and σ² = np(1-p), for the binomial di

In this post, we introduce the basics of hypothesis testing using the binomial distribution  — a key topic in A-Level Maths (Year 12 Statistics) . You’ll learn how to structure hypotheses, decide which tail to check, and determine whether results are statistically significant. What is Hypothesis Testing? Hypothesis testing is a statistical method used to decide whether an observed result provides sufficient evidence to challenge an initial assumption. For example, suppose you

Introduction This post explains how to find the general solutions of trigonometric equations in radians . The method is identical to the degrees version: find the principal value  and then use the appropriate general solution formula  for sine, cosine or tangent. These examples are ideal for A-level Maths and Further Maths students who need to work confidently in radians. Remember: π radians = 180° In radian mode  your calculator can give solutions either as decimals or in te

This guide explains how to solve general solutions of trigonometric equations in degrees  for sine, cosine, and tangent, using the PV method, with step-by-step examples suitable for A-level and Further Maths GCSE students. How to Solve General Trigonometric Equations in Degrees Trigonometric functions repeat at regular intervals, so there’s usually more than one solution. The Principal Value (PV) method  is a simple and visual way to find all solutions. Steps: Find the Princi

There are two key concepts you need to grasp: standard integrals and generalised integrals. Standard integrals, also known as elementary integrals, form the backbone of calculus. They encompass well-known integration formulas used to integrate specific functions or types of functions. Generalised integrals expand our integration toolkit to encompass a broader range of functions. By introducing parameters like a and b , we can adapt standard integration formulas to handle mor