Introduction to Probability (IGCSE 0607)

Hello mathematicians! Welcome to the exciting world of Probability. This chapter is all about understanding chance and predicting how likely events are to happen.

Why is this important? Probability helps us make decisions every day—from choosing whether to take an umbrella (what is the chance of rain?) to assessing risk in finance or science. It’s a core skill for understanding data and statistics.
We will start with the basics (Core content) and move into more powerful techniques for handling combined events (Extended content). Don't worry if this seems tricky at first; we will break down every concept step-by-step!

Section 1: The Probability Scale and Single Events (C9.1 / E9.1)

Probability is simply a measure of how likely something is to happen.

1.1 The Probability Scale

All probabilities live on a scale from 0 to 1.

  • 0: The event is impossible. (e.g., The probability of a cat growing wings.)
  • 0.5 or 1/2: The event is even chance. (e.g., The probability of flipping a head.)
  • 1: The event is certain. (e.g., The probability that the sun will rise tomorrow.)

Probabilities must be given as a fraction, decimal, or percentage (e.g., 1/4, 0.25, or 25%).

1.2 Calculating the Probability of a Single Event

The core formula for calculating the probability of a specific event (\(A\)) is straightforward:

$$P(A) = \frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}}$$

Example: Rolling a Die

Imagine you roll a fair, six-sided die.

  • Total possible outcomes: {1, 2, 3, 4, 5, 6}. Total = 6.
  • Event A: Rolling a 4.
    Favourable outcomes: {4}. Number = 1.
    $$P(\text{Rolling a 4}) = \frac{1}{6}$$
  • Event B: Rolling an even number.
    Favourable outcomes: {2, 4, 6}. Number = 3.
    $$P(\text{Rolling an even number}) = \frac{3}{6} = \frac{1}{2}$$

1.3 Complementary Events (The 'Not' Rule)

The probability of an event happening plus the probability of that event not happening must always equal 1. These are called complementary events.

$$P(\text{Not } A) = 1 - P(A)$$

If we use the Extended syllabus notation (which is not required for Core candidates, but good to know!): \(P(A')\) or \(P(\bar{A})\) means the probability of A not occurring.
$$P(A') = 1 - P(A)$$

Quick Review: Single Events

The probability scale goes from 0 to 1. Always check if the question asks for the probability of the event happening or not happening.

Section 2: Relative Frequency and Expected Frequency (C9.2 / E9.2)

Sometimes, we can’t calculate probability theoretically (like rolling a die). Instead, we have to rely on experiments. This leads us to frequency.

2.1 Relative Frequency (Experimental Probability)

The Relative Frequency is the probability estimated from carrying out an experiment.
It is sometimes called Experimental Probability.

$$\text{Relative Frequency} = \frac{\text{Number of times the event occurred}}{\text{Total number of trials}}$$

Did you know?

The more times you repeat an experiment (the greater the number of trials), the closer the relative frequency usually gets to the true theoretical probability. This is called the Law of Large Numbers.

Example: Spinning a Spinner

A spinner is spun 200 times. It lands on Red 45 times.
The estimated probability \(P(\text{Red})\) is the relative frequency:
$$P(\text{Red}) = \frac{45}{200} = 0.225$$

2.2 Understanding Fairness and Bias

When dealing with probability, we often talk about:

  • Fair/Random: Every outcome has an equal chance of occurring. (e.g., a balanced coin, a standard die.)
  • Bias: Some outcomes are more likely than others. This is why we rely on relative frequency to estimate the probability of biased objects. (e.g., a weighted coin.)

2.3 Calculating Expected Frequency

If we know the probability of an event, we can predict how many times it will occur in a set number of future trials. This is the Expected Frequency (or expected value).

$$\text{Expected Frequency} = P(\text{Event}) \times \text{Total Number of Trials}$$

Example: Expected Wins

The probability that a football team wins a match is 0.6. If they play 40 matches this season, how many wins are expected?
$$\text{Expected Wins} = 0.6 \times 40 = 24$$

Key Takeaway: Estimation

Relative frequency is what actually happened in the past. Expected frequency is what we predict will happen in the future.

Section 3: Probability of Combined Events (C9.3 / E9.3)

When two or more events happen together (e.g., flipping two coins, drawing two cards), we look at combined probability. We use diagrams to visualize all possibilities.

3.1 Sample Space Diagrams

A Sample Space Diagram (often a list or a two-way table) shows all the possible outcomes in an experiment.

Example: Rolling two dice (Two-way table)

When rolling two dice, the sample space has \(6 \times 6 = 36\) possible outcomes.
If we want the probability of getting a total sum of 7:

  • Favourable outcomes (sums of 7): (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). Total = 6.
  • $$P(\text{Sum is 7}) = \frac{6}{36} = \frac{1}{6}$$

3.2 Using Venn Diagrams

Venn diagrams are great for visualizing events and their relationships, especially where two sets overlap (Core and Extended syllabus limits Venn diagrams to two sets).

  • The rectangle represents the Universal Set (\(U\)), or all possible outcomes.
  • Circles represent the specific events (\(A\), \(B\)).
Key Terms for Venn Diagrams (C9/E9)
  • Intersection (AND): The middle part where \(A\) and \(B\) overlap. This means both events happen. Notation (Extended): \(A \cap B\).
  • Union (OR): The entire area covered by \(A\) or \(B\). This means A happens, or B happens, or both happen. Notation (Extended): \(A \cup B\).
  • Complement (NOT): The area outside the circle(s). Notation (Extended): \(A'\).

3.3 Tree Diagrams

Tree diagrams are essential for showing sequences of two or more events.

How to use a Tree Diagram:

  1. Start with the first event and draw branches for all its possible outcomes, writing the probability on the branch.
  2. From the end of those branches, draw secondary branches for the second event, again writing the probabilities on the new branches.
  3. Write the final outcome (e.g., "Head, Tail") and the final probability at the end of each path.

The Golden Rules of Tree Diagrams:

  • Rule 1 (Along the branches): To find the probability of a sequence (e.g., Red AND Red), you multiply the probabilities along the path.
  • Rule 2 (Down the end columns): To find the probability of multiple successful sequences (e.g., Red, Blue OR Blue, Red), you add the probabilities of the required end points.
Common Mistake Alert!

Don't confuse adding and multiplying! You multiply the probabilities along the branches (AND) and add the probabilities at the end (OR).

Section 4: Advanced Combined Probability (Extended Focus E9.3)

For Extended students, you need to formalize the rules for combining events and consider the important difference between events with replacement and those without.

4.1 Independent Events (With Replacement)

Two events, \(A\) and \(B\), are Independent if the outcome of one event does not affect the probability of the other event.

Analogy: Flipping a coin twice. The result of the first flip has no impact on the second flip.

Multiplication Rule for Independent Events (AND):
$$P(A \text{ and } B) = P(A) \times P(B)$$

Events with Replacement

When items are selected and then put back (replaced), the events are independent because the total pool of items remains the same for the next pick.

4.2 Dependent Events (Without Replacement)

Two events are Dependent if the outcome of the first event does change the probability of the second event.

Analogy: Drawing marbles from a bag without replacing the first one. If you remove a red marble, the probability of drawing another red marble changes for the second draw.

In tree diagrams for dependent events, you must carefully update the probabilities for the second set of branches based on what happened in the first event!

Example: A bag has 3 Red and 7 Blue marbles.

  • \(P(\text{1st pick is Red}) = \frac{3}{10}\)
  • If the first was Red (and not replaced), the bag now has 2 Red and 7 Blue (Total 9).
  • \(P(\text{2nd pick is Red}) = \frac{2}{9}\)
  • $$P(\text{Red and then Red}) = \frac{3}{10} \times \frac{2}{9} = \frac{6}{90}$$

4.3 Mutually Exclusive Events (The Addition Rule)

Two events, \(A\) and \(B\), are Mutually Exclusive if they cannot happen at the same time (they have no outcomes in common). If \(A\) happens, \(B\) cannot, and vice versa.

Analogy: Rolling a die. You cannot roll a 2 AND a 5 at the same time.

Addition Rule for Mutually Exclusive Events (OR):
$$P(A \text{ or } B) = P(A) + P(B)$$

The syllabus uses the notation: $$P(A \cup B) = P(A) + P(B)$$

Memory Aid: When to Add, When to Multiply

Adding = M.E. (Mutually Exclusive). Use + when you want one outcome OR another.
Multiplying = Independent. Use x when you want one outcome AND another (a sequence).

Review and Practice: Probability Tools

Use the correct diagram based on the question structure:

Tool Summary
  • Sample Space Diagram / Table: Best for visualizing all possible outcomes, especially when dealing with two independent events (like rolling two dice or flipping two coins).
  • Venn Diagram: Best for visualizing outcomes based on shared characteristics or sets (e.g., people who like tea AND coffee).
  • Tree Diagram: Essential for sequences of events, particularly when the events are dependent (without replacement).

You've covered the foundation of probability! The key to mastery is practice, focusing on whether events are independent or mutually exclusive, and choosing the right tool (table, Venn, or tree) to calculate the final probability. Keep up the great work!