Introduction to Probability

Welcome to the world of Probability! At its heart, probability is simply the mathematics of uncertainty. Whether you are wondering if it will rain tomorrow, predicting the winner of a football match, or rolling dice in a board game, you are using probability. In this chapter, we will learn the language used to describe how likely things are to happen and the basic tools you need to calculate these chances.

Don't worry if you find math a bit "hit or miss" sometimes—probability is one of the most practical parts of the Pearson Edexcel Specification B course, and once you master the basic vocabulary, the calculations are very straightforward!

1. The Probability Scale

In everyday life, we use words like "maybe," "definitely," or "no way." In mathematics, we turn these words into a precise scale. Probability is always measured on a scale from \(0\) to \(1\).

  • \(0\) (Impossible): The event cannot happen (e.g., rolling a \(7\) on a standard six-sided die).
  • \(1\) (Certain): The event will definitely happen (e.g., the sun rising tomorrow).
  • \(0.5\) (Even Chance): The event is just as likely to happen as not to happen (e.g., getting "Heads" on a fair coin).

Important Tip: You can write probabilities as fractions, decimals, or percentages. For example, a half-chance can be written as \(\frac{1}{2}\), \(0.5\), or \(50\%\). However, in your IGCSE exam, fractions and decimals are most common.

Common Mistake to Avoid: Never give a probability answer greater than \(1\) or less than \(0\). If you calculate a probability of \(1.2\), something has gone wrong!

2. Sample Spaces

A Sample Space is simply a fancy name for "a list of every possible outcome" for an experiment. If we know all the possible results, we can calculate how likely one specific result is.

Example: Rolling a fair six-sided die
The sample space is: \(\{1, 2, 3, 4, 5, 6\}\).
The number of elements in this set is \(n(S) = 6\).

Example: Tossing a coin
The sample space is: \(\{\text{Heads, Tails}\}\).
The number of elements is \(n(S) = 2\).

Quick Review:

In Specification B, we often use set notation. If \(S\) is the sample space, then \(n(S)\) represents the total number of possible outcomes.

3. Calculating Basic Probability

If all outcomes in a sample space are equally likely (like a fair die or a fair coin), we use a simple formula to find the probability of an event, which we call \(P(\text{Event})\):

\(P(\text{Event}) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\)

Example: What is the probability of rolling an even number on a standard die?
1. Successful outcomes (even numbers): \(\{2, 4, 6\}\). There are \(3\) of these.
2. Total possible outcomes: \(\{1, 2, 3, 4, 5, 6\}\). There are \(6\) of these.
3. \(P(\text{Even}) = \frac{3}{6} = \frac{1}{2}\) (or \(0.5\)).

4. Complementary Events

In probability, the complement of an event is the event not happening. Using the language of Sets (which you might remember from Section 2 of the syllabus), if an event is \(A\), its complement is written as \(A'\).

Because it is certain that an event either happens or doesn't happen, the sum of their probabilities is always \(1\):

\(P(A) + P(A') = 1\)

This leads to a very useful formula for when you want to find the chance of something not happening:

\(P(\text{not } A) = 1 - P(A)\)

Example: If the probability that it will rain tomorrow is \(0.3\), what is the probability that it will not rain?
\(P(\text{not rain}) = 1 - 0.3 = 0.7\).

Key Takeaway: If a question asks for the probability of "at least one" or "not something," it is often easier to find the probability of the thing you don't want and subtract it from \(1\).

5. Relative Frequency (Experimental Probability)

Sometimes we don't have a "fair" object like a die. For example, if we want to know the probability of a drawing pin landing point-up, we can't just guess. We have to perform an experiment.

Relative Frequency is the probability calculated from an experiment or from data. The formula is:

\(\text{Relative Frequency} = \frac{\text{Number of times the event happened}}{\text{Total number of trials}}\)

Did you know?

The more times you repeat an experiment (the more trials you do), the more reliable your relative frequency becomes. If you flip a coin \(10\) times, you might get \(7\) heads by accident. If you flip it \(1,000\) times, you are much more likely to get a relative frequency close to \(0.5\).

Example: A bag contains an unknown number of colored beads. A student picks a bead, records the color, and replaces it. They do this \(50\) times. If they pick a blue bead \(12\) times, what is the relative frequency of picking a blue bead?
\(\text{Relative Frequency} = \frac{12}{50} = 0.24\).

Summary of Key Concepts

1. The Scale: Probabilities are between \(0\) and \(1\).
2. Sample Space: The list of all possible outcomes.
3. Theoretical Probability: Used when outcomes are equally likely (\(\frac{\text{want}}{\text{total}}\)).
4. Complements: The probability of something NOT happening is \(1 - P(\text{happening})\).
5. Relative Frequency: Probability based on real-world trials or experiments.

Note: For more complex problems involving multiple events, see the upcoming chapters on "Addition and product rules" and "Tree diagrams."