Probability: Single & Combined Events

1. The Core Principle of Probability

At its heart, probability is a way of measuring the likelihood of an event happening. It's always a value between 0 (impossible) and 1 (certain). For your exams, you'll almost always express probabilities as fractions.

The fundamental formula is:
$P(\text{event})=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

A crucial related idea is the probability of an event not happening. Since the total probability of all possible outcomes must be 1, we can say:
$P(\text{not A})=1-P(\text{A})$

Let's watch a very quick example that puts these two simple ideas into the context of an IGCSE question.

Expected Frequency

A common application of single event probability is calculating the expected frequency (or expected number of outcomes). If you know the probability of an event and the number of times you're going to run the experiment (the number of trials), you can predict how many times you expect that outcome to occur.

The formula is straightforward:
Expected Frequency = Probability of the event × Number of trials

This next clip from the same video provides a clear, worked example.

2. Combined Events: The 'AND' and 'OR' Rules

Most exam questions involve more than one event happening in sequence. These are called combined events. To solve these, you need to understand two key rules: the 'AND' rule and the 'OR' rule.

First, we must distinguish between two types of combined events:

• Independent Events: The outcome of the first event does not affect the probability of the second. A classic example is picking an item and then replacing it.
• Dependent Events: The outcome of the first event does affect the probability of the second. This happens when you pick an item without replacement.

The 'AND' Rule (Multiplication)

When you want to find the probability of event A and then event B happening, you multiply their probabilities.
$P(\text{A and B})=P(\text{A})\times P(\text{B})$

• For independent events, $P(\text{B})$ is its standard probability.
• For dependent events, $P(\text{B})$ is the new probability, after event A has already occurred.

The 'OR' Rule (Addition)

When you want to find the probability of event A or event B happening (where A and B cannot happen at the same time), you add their probabilities.
$P(\text{A or B})=P(\text{A})+P(\text{B})$

This is most often used when a question asks for a final outcome that can be achieved in multiple ways. For example, "picking two balls of the same colour" could mean (Red and Red) OR (Blue and Blue).

Let's watch a video that works through examples of these rules for both independent (with replacement) and dependent (without replacement) events.

To solidify your understanding, take a few minutes to read through this summary page.

3. Structuring Your Work: Tree Diagrams

Tree diagrams are an invaluable tool for organising your thoughts in probability questions. They provide a clear visual map of all possible outcomes and their probabilities, which is essential for avoiding the small arithmetic errors you've mentioned can be a problem.

The rules for using a tree diagram are a direct application of the AND/OR rules:

1. Multiply along the branches to find the probability of a specific sequence of outcomes (the 'AND' rule).
2. Add the results at the end for the different outcome paths that satisfy the question (the 'OR' rule).

This revision note gives a quick, effective overview of how to set up and use a tree diagram.

Test your understanding!

The image below shows a probability problem. Before looking at the solution in the image, try to solve it yourself.
Question: Anna has a bag with 2 red and 3 yellow sweets. Ben has a bag with 4 red and 1 yellow sweet. Anna takes a sweet, then Ben takes a sweet. Calculate the probability they both take a red sweet.

Show answer

This is a case of two independent events, as the contents of Ben's bag are not affected by Anna's choice.

1. Probability of Anna picking red:
   $P(\text{Anna picks Red})=\frac{2}{5}$ (2 red out of 5 total)

2. Probability of Ben picking red:
   $P(\text{Ben picks Red})=\frac{4}{5}$ (4 red out of 5 total)

3. Apply the 'AND' Rule:
   We need Anna to pick red AND Ben to pick red.
   $P(\text{Both pick Red})=P(\text{Anna Red})\times P(\text{Ben Red})=\frac{2}{5}\times \frac{4}{5}=\frac{8}{25}$
   As you can see in the image, this matches the result found using the tree diagram.

4. Key Exam Strategies and Scenarios

To move from a B to an A grade, you need to master not just the basics, but also the common traps and efficient methods that save time in an exam.

Strategy 1: The "At Least One" Shortcut

If a question asks for the probability of "at least one" of something, it's often much faster to calculate the probability of "none" and subtract it from 1.
$P(\text{at least one})=1-P(\text{none})$

For example, $P(\text{at least one red ball})$ is the same as $1-P(\text{no red balls})$. This saves you from calculating $P(\text{1 red})+P(\text{2 reds})+P(\text{3 reds})$ etc.

Watch this video to see a side-by-side comparison of the long method versus this shortcut.

Strategy 2: Dealing with Unspecified Order

A common exam pitfall is when the question asks for a combination of outcomes but doesn't specify the order. For example, "find the probability of picking one red and one blue sweet".

Many students only calculate P(Red then Blue) and forget that P(Blue then Red) also satisfies the condition. You must calculate the probability for all possible permutations and add them together.

The video below works through a problem of picking 3 pens (2 blue, 1 red) without replacement. The presenter clearly shows the three possible permutations and a clever shortcut.

Strategy 3: Tackling Probability "Show That" Questions

You've identified "show that" questions as a weakness. In probability, these often involve forming an algebraic expression for a probability and showing it simplifies to a given quadratic equation.

The key is to stay calm and build the probability expression step-by-step. Don't panic if your algebra doesn't immediately match the target; instead, use the target expression to help find your mistake.

This final video is an advanced, high-mark exam question that brings all these ideas together. It is an excellent test of your understanding.

Conclusion

Excellent work today. We've moved from the basic definition of probability right up to the complex, multi-step problems you're likely to see at the top end of your exam papers.

Key Takeaways:

• Single Events: Probability is the ratio of favorable outcomes to total outcomes. Expected frequency is this probability multiplied by the number of trials.
• Combined Events: Use the 'AND' rule (multiply) for events in sequence and the 'OR' rule (add) for mutually exclusive options.
• Dependent vs. Independent: Always check if events are with replacement (independent) or without replacement (dependent), as this changes the probabilities for subsequent events.
• Exam Strategy: Use tree diagrams to structure your work, use the $1-P(\text{none})$ shortcut for "at least one" problems, and always account for all permutations when the order is not specified.

Preview of the Next Lesson:
In the next lesson, we will continue our work on probability by learning how to use Venn diagrams. This is another visual tool that helps solve problems involving overlapping sets and introduces the important concept of conditional probability in a different context.