Welcome to Probability: Understanding Chance!

Hello! In this chapter, we dive into the world of Probability, which is simply the mathematics of chance. Probability helps us measure how likely an event is to happen. Whether you are checking the weather forecast, predicting the outcome of a game, or making business decisions, probability is a fundamental skill used every day.

Don't worry if this seems tricky at first! We will break down every concept step-by-step, starting with the very basics of measuring uncertainty. Let's make sure you feel confident calculating chances!

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

The Probability Scale

Probability is always measured on a scale from 0 to 1.

  • 0: The event is impossible (e.g., the probability of the sun rising in the west).
  • 0.5 or 50%: The event has an even chance (e.g., flipping a fair coin and getting heads).
  • 1: The event is certain (e.g., the probability of a human needing to breathe).

Probabilities can be expressed in three ways, and you must be comfortable converting between them:

  1. Fractions (e.g., \(\frac{1}{4}\))
  2. Decimals (e.g., 0.25)
  3. Percentages (e.g., 25%)

Calculating the Probability of a Single Event

The probability of an event happening is calculated using the following formula:

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

Example: A bag contains 3 red balls, 5 blue balls, and 2 yellow balls. What is the probability of picking a blue ball?

  • Total possible outcomes = \(3 + 5 + 2 = 10\)
  • Favourable outcomes (Blue balls) = 5
  • Probability of Blue = \(\frac{5}{10} = \frac{1}{2}\) or 0.5 or 50%.

Complementary Events

The probability of an event not happening is called its complement. Since probability must add up to 1 (or 100%), we use a simple rule:

The probability of an event not occurring = \(1\) – the probability of the event occurring.

If Event A is rolling a 6 on a dice, then Not A (or \(A'\)) is rolling a 1, 2, 3, 4, or 5.

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

Example (from syllabus): The probability that a counter is blue is 0.8. What is the probability that it is not blue?

\(P(\text{not blue}) = 1 - P(\text{blue}) = 1 - 0.8 = 0.2\).

(Extended Note: The notation used for the complement of event A is \(P(A')\). So, \(P(A') = 1 - P(A)\).)

Key Takeaway (Section 1)

Probability is a ratio from 0 to 1. Always ensure your final probability is simplified (if a fraction) or in the required format (decimal or percentage). The chance of something NOT happening is 1 minus the chance it WILL happen.

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

What is Relative Frequency?

In Section 1, we dealt with Theoretical Probability (what should happen in a perfect world). When we perform an experiment, we find the Relative Frequency (also known as Experimental Probability), which is what actually happened.

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

The syllabus requires you to understand relative frequency as an estimate of probability. The more times you repeat an experiment, the closer the relative frequency gets to the true theoretical probability.

Fair, Bias, and Random

These terms describe the nature of an event or experiment:

  • Random: All outcomes have an equal chance of happening (e.g., drawing a name out of a hat blindfolded).
  • Fair: Used when describing equipment. A fair coin or die means the theoretical probability for each outcome is equal (e.g., \(P(\text{Heads}) = 0.5\)).
  • Bias: Used when equipment is unfair. If you roll a die 100 times and get a '6' fifty times, you suspect the die is biased, and the relative frequency (0.5) is a better estimate of \(P(6)\) than the theoretical probability (\(\frac{1}{6}\)).

Calculating Expected Frequencies

If you know the probability of an event, you can estimate how many times it will happen over a large number of trials. This is the Expected Frequency.

Expected Frequency = \(P(\text{Event}) \times \text{Number of Trials (or population size)}\)

Example: The probability of a train being late is 0.15. If 300 trains run next week, how many trains are expected to be late?

Expected late trains = \(0.15 \times 300 = 45\) trains.

Did you know? Probability is essential for quality control in manufacturing. If a manufacturer knows 1% of products are faulty, they can use expected frequency to estimate how many faulty items they will produce each month.

Quick Review (Section 2)

Relative Frequency is based on experiments. Expected Frequency estimates results for a large group using the calculated probability.

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

When two or more events happen together (or one after the other), we are dealing with Combined Events. We use diagrams to organize this information.

Method 1: Sample Space Diagrams

These are diagrams or tables listing all the possible outcomes when two events occur. They are usually used for simple, simultaneous events, like rolling two dice or flipping two coins.

Example: Rolling two fair six-sided dice and summing the scores.

(Imagine a table here with 6 rows and 6 columns, showing all 36 possible outcomes.)

The total number of outcomes (the sample space) is \(6 \times 6 = 36\).
To find the probability of getting a sum of 7, we count the ways: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1). There are 6 ways.
\(P(\text{Sum of 7}) = \frac{6}{36} = \frac{1}{6}\).

Method 2: Venn Diagrams (Limited to Two Sets)

Venn diagrams are great for visualizing how outcomes overlap or belong to different categories.

The syllabus limits us to problems involving two sets.

  • The rectangle represents the Universal Set (\(U\))—all possible outcomes.
  • Circles represent specific events (Sets A and B).
Notation for Venn Diagrams (Extended Only)

For Core students, you will use "and" or "or". For Extended students, you need the official notation:

  • Intersection (\(\cap\)): \(P(A \cap B)\) means the probability of A AND B happening (the overlap).
  • Union (\(\cup\)): \(P(A \cup B)\) means the probability of A OR B (or both) happening (the whole area covered by A and B).

Analogy: Think of two circles. The Union (\(\cup\)) is everything inside Circle A OR Circle B. The Intersection (\(\cap\)) is the small section where they overlap.

Method 3: Tree Diagrams

Tree diagrams are used for sequential events (one event followed by another).

Step-by-Step Guide for Tree Diagrams
  1. Draw Branches: Draw branches for the first event, then from the end of those branches, draw branches for the second event.
  2. Write Probabilities: Write the probability on each branch by the side.
  3. Calculate Outcomes (Multiply): To find the probability of a sequence (e.g., Red then Blue), multiply the probabilities along the required branches (this represents the "AND" rule). Outcomes are written at the end of the branches.
  4. Combine Outcomes (Add): If the question asks for multiple ways an event can happen (e.g., one Red and one Blue), you calculate the probabilities of each path and add them together (this represents the "OR" rule).
Mutually Exclusive vs. Independent Events (Extended Only)
Mutually Exclusive Events

These are events that cannot happen at the same time. If you roll a die, getting a 5 and getting an even number are mutually exclusive.

Addition Rule (OR): \(P(A \text{ or } B) = P(A) + P(B)\)
(Note: Since they cannot overlap, \(P(A \cap B) = 0\)).

Independent Events

These are events where the outcome of the first event does not affect the outcome of the second event. (e.g., flipping a coin twice).

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

Example of Independent Events: You flip a coin and roll a die.
\(P(\text{Heads and a 4}) = P(\text{H}) \times P(\text{4}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}\).

Combined Events with and Without Replacement (Extended Only)

This is a key concept often tested using tree diagrams.

  • With Replacement: The first item is put back before the second draw. The events are independent, and the probabilities on the second set of branches remain the same.
  • Without Replacement: The first item is NOT put back. The events are dependent, and the probabilities on the second set of branches must change because the total number of items (and potentially the number of favourable outcomes) has decreased by one.

Tricky Tip: When doing 'without replacement' problems, always check the total number of remaining items and the number of specific items left before you write the probability on the second set of branches!

Key Takeaway (Section 3)

Use Sample Space Diagrams for simple simultaneous events. Use Tree Diagrams for sequential events, remembering to multiply along branches and add paths. If the event is 'without replacement' (Extended), probabilities change!

Final Summary Check

  • The probability scale goes from 0 to 1.
  • \(P(\text{Event}) = \frac{\text{Favourable}}{\text{Total}}\)
  • \(P(\text{not } A) = 1 - P(A)\)
  • Expected frequency is calculated by multiplying probability by the number of trials.
  • (Extended) Mutually Exclusive (OR): \(P(A) + P(B)\)
  • (Extended) Independent (AND): \(P(A) \times P(B)\)

You've mastered the fundamentals of chance! Keep practicing those Venn and Tree diagrams, as they are crucial for solving complex probability questions. Good luck!