Probability in O-Level Maths: Tree Diagrams, Tables and Conditional Cases

Probability is one of the most consistently tested topics in the Singapore O-Level E-Mathematics (Syllabus 4052) examination, and for good reason — it combines logical thinking, systematic working, and arithmetic accuracy all at once. Yet many students lose marks not because the concepts are impossibly hard, but because they are unsure which method to use and when. Should you draw a tree diagram or construct a table? Should you multiply probabilities or add them? What exactly does a “given that” question want from you?

This guide breaks down every key probability method tested at the O-Level: possibility tables (sample space diagrams), tree diagrams for both independent and dependent events, and conditional probability. Each section includes clear explanations and worked examples so you can move from understanding the concept to applying it confidently in an exam. Whether you are just starting out or doing a final revision sweep, this is your go-to reference for probability in O-Level Maths.

What Is Probability and Why Does It Matter in O-Level Maths?

Probability measures how likely an event is to occur, expressed as a value between 0 (impossible) and 1 (certain). In everyday terms, it answers questions like: “If I draw a card at random, how likely am I to pick a heart?” or “If I roll two dice, what are the chances they sum to 7?” In the O-Level context, probability questions typically involve two or more events happening in sequence, and your job is to find the probability of a specific combined outcome.

The O-Level E-Maths syllabus requires students to calculate probabilities of both single and combined events, and to represent the sample space using appropriate tools. The three main tools you need to master are possibility tables, tree diagrams, and the conditional probability formula. Understanding when to use each one is just as important as knowing how to use it.

Key Concepts: Independent vs Dependent Events

Before picking up your pencil to draw any diagram, you need to identify what type of events you are dealing with. This single determination shapes everything that follows.

Independent Events are events where the outcome of one does not affect the outcome of the other. The classic example is drawing a ball from a bag with replacement — because the ball is returned before the next draw, the total number of balls in the bag stays the same, and the probabilities for the second draw are identical to the first. Tossing a coin twice is another example: the result of the first toss has absolutely no bearing on the second.

Dependent Events are events where the outcome of one directly influences the probability of the next. Drawing a ball from a bag without replacement is the textbook example: once the first ball is removed, the bag now contains one fewer ball, so the probabilities for the second draw change. In real O-Level questions, dependent events almost always involve the phrase “without replacement” or a scenario where a person or item is not returned to the group.

Keeping this distinction clear in your mind from the very start of a question will prevent the most common errors students make when setting up their diagrams and calculations.

Possibility Tables (Sample Space Diagrams)

A possibility table — also called a sample space diagram — is a grid that systematically lists every possible outcome when two events are combined. One event’s outcomes run along the top of the table (columns) and the other event’s outcomes run down the side (rows). Each cell of the table then records the combined outcome for that pair.

Possibility tables are most useful when both events have multiple outcomes and the combined outcomes can be expressed as a number (such as a sum, difference, or product). They give you a clear visual picture of the entire sample space, making it easy to count favourable outcomes and calculate probabilities accurately.

How to Construct a Possibility Table

1. Identify both events and their individual outcomes. For example, rolling two dice means each die has outcomes 1, 2, 3, 4, 5, 6.
2. Set up the grid. List the outcomes of the first event across the top row and the outcomes of the second event down the left column.
3. Fill in every cell by combining the row value and the column value according to the operation required (sum, product, difference, etc.).
4. Count the total number of cells — this is your total sample space, n(S).
5. Highlight or circle the favourable outcomes (those that satisfy the condition in the question) and count them to get n(A).
6. Calculate the probability: P(A) = n(A) ÷ n(S).

Worked Example: Rolling Two Dice

Question: Two fair six-sided dice are rolled. Using a possibility table, find the probability that the sum of the two numbers is (a) equal to 7, and (b) greater than 9.

Setting up the table with Die 1 outcomes (1–6) along the top and Die 2 outcomes (1–6) down the side, and filling each cell with the sum of the two dice, gives a 6×6 grid with 36 total outcomes.

(a) Sum = 7: The combinations that give a sum of 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1) — that is 6 favourable outcomes.
P(sum = 7) = 6/36 = 1/6

(b) Sum > 9: Sums of 10, 11, and 12 can be found in the table. The combinations are (4,6), (5,5), (6,4) for sum 10; (5,6), (6,5) for sum 11; and (6,6) for sum 12 — giving 6 favourable outcomes.
P(sum > 9) = 6/36 = 1/6

Notice that the possibility table makes it very easy to count outcomes without missing any. Attempting to list these mentally would be risky in an exam setting.

Probability Tree Diagrams

A probability tree diagram is a branching visual that maps out every possible sequence of outcomes for two or more successive events. The diagram starts at a single root point and branches outward — each branch represents one possible outcome of an event, and the probability of that outcome is written along the branch. Outcomes appear at the end of each branch.

Tree diagrams are particularly powerful when events have fewer possible outcomes each (such as drawing a ball that is either red or blue) but are being combined across multiple trials. They are also the natural tool for dependent events, because the changing probabilities after each event can be shown clearly on each new set of branches. One key rule applies to every tree diagram: the probabilities on all branches from any single point must always add up to 1.

How to Draw a Tree Diagram

1. Draw the first set of branches from the root point. Label each branch with its outcome and write the probability above or below the branch.
2. From the end of each first-level branch, draw the second set of branches. Label outcomes and probabilities — remembering that if events are dependent, these probabilities may differ from the first level.
3. Verify each set of branches sums to 1 before proceeding. This is a quick error-check.
4. To find the probability of any complete path (sequence of outcomes), multiply the probabilities along that path.
5. To find the probability of a combined event that can occur via multiple paths, add the probabilities of all relevant paths.

Tree Diagrams for Independent Events (With Replacement)

Question: A bag contains 3 red balls and 2 blue balls. A ball is drawn at random, its colour noted, and then returned to the bag. A second ball is then drawn. Find the probability that (a) both balls are red, and (b) the two balls are different colours.

Because the first ball is replaced, the total in the bag remains 5 for both draws. This means the second-level branches are identical to the first-level branches, confirming these are independent events.

• P(Red on any draw) = 3/5
• P(Blue on any draw) = 2/5

Drawing the tree diagram, the four complete paths and their probabilities are:

• Red then Red: 3/5 × 3/5 = 9/25
• Red then Blue: 3/5 × 2/5 = 6/25
• Blue then Red: 2/5 × 3/5 = 6/25
• Blue then Blue: 2/5 × 2/5 = 4/25

(a) P(both red) = 9/25

(b) P(different colours) = P(Red then Blue) + P(Blue then Red) = 6/25 + 6/25 = 12/25

As a final check, all four path probabilities should sum to 1: 9/25 + 6/25 + 6/25 + 4/25 = 25/25 ✓

Tree Diagrams for Dependent Events (Without Replacement)

Question: Using the same bag (3 red balls, 2 blue balls), a ball is drawn and not replaced before a second ball is drawn. Find the probability that (a) both balls are red, and (b) exactly one ball is red.

Because the first ball is not returned, the bag now contains only 4 balls for the second draw. The second-level probabilities change depending on what colour was drawn first — this is what makes them dependent events.

If the first ball drawn is Red (probability 3/5), the bag now has 2 red and 2 blue balls remaining (4 total):

• P(Red | first was Red) = 2/4 = 1/2
• P(Blue | first was Red) = 2/4 = 1/2

If the first ball drawn is Blue (probability 2/5), the bag now has 3 red and 1 blue ball remaining (4 total):

• P(Red | first was Blue) = 3/4
• P(Blue | first was Blue) = 1/4

The complete path probabilities are:

• Red then Red: 3/5 × 1/2 = 3/10
• Red then Blue: 3/5 × 1/2 = 3/10
• Blue then Red: 2/5 × 3/4 = 6/20 = 3/10
• Blue then Blue: 2/5 × 1/4 = 2/20 = 1/10

(a) P(both red) = 3/10

(b) P(exactly one red) = P(Red then Blue) + P(Blue then Red) = 3/10 + 3/10 = 3/5

Check: 3/10 + 3/10 + 3/10 + 1/10 = 10/10 = 1 ✓

When to Multiply and When to Add Probabilities

One of the most common points of confusion in probability is knowing whether to multiply or add. The good news is that this follows two clear rules, and once you understand the logic behind them, they become second nature.

Multiply probabilities when events happen together (AND). If you want the probability of outcome A and then outcome B occurring in the same sequence, you multiply along the branches of your tree diagram. This is because both things need to happen, and the probability of a combined sequence is always smaller than either individual probability.

Add probabilities when you are combining separate, mutually exclusive paths (OR). If there are two or more different ways an event can occur — for example, getting exactly one red ball could happen via “Red then Blue” OR “Blue then Red” — you add the probabilities of those separate paths. Mutually exclusive outcomes cannot happen at the same time, so adding them gives the total probability of any one of them occurring.

A quick mental check: if your answer after multiplying seems too high (greater than either individual probability), something has gone wrong. Similarly, if after adding you get a value greater than 1, you have likely added paths that are not mutually exclusive.

Conditional Probability: The “Given That” Questions

Conditional probability is the probability of one event occurring given that another event has already happened or is known to have occurred. These questions are often the highest-difficulty items in O-Level probability, and they are signalled by the phrase “given that” in the question. For example: “Given that the first ball drawn is red, find the probability that the second ball is also red.”

The key insight in conditional probability is that knowing one event has occurred restricts your sample space. You are no longer looking at all possible outcomes — only the outcomes that are consistent with the condition that has been given. This is what makes conditional probability feel different from regular combined probability, and it is why students sometimes stumble if they approach these questions without a clear method.

The Conditional Probability Formula

The formal notation for conditional probability is P(A | B), read as “the probability of A given B.” The formula is:

P(A | B) = P(A ∩ B) / P(B)

Where P(A ∩ B) is the probability that both A and B occur together, and P(B) is the probability that event B occurs (and must not be zero). This formula effectively scales the probability of both events occurring by the probability of the condition, giving you the updated likelihood of A within the restricted sample space defined by B.

In many O-Level questions, you can also read conditional probabilities directly off a tree diagram or a table — the formula becomes most useful when you are given summary probabilities rather than a full diagram.

Worked Example: Conditional Probability

Question: In a class of 30 students, 18 study Mathematics, 12 study Physics, and 7 study both subjects. A student is chosen at random. Given that the student studies Physics, find the probability that they also study Mathematics.

Here, event A is “studies Mathematics” and event B is “studies Physics.” The condition given is that the student studies Physics, so we restrict our focus to only the 12 Physics students.

• P(B) = P(studies Physics) = 12/30 = 2/5
• P(A ∩ B) = P(studies both) = 7/30

Applying the formula:

P(A | B) = P(A ∩ B) / P(B) = (7/30) ÷ (12/30) = 7/12

Answer: The probability that the student studies Mathematics, given that they study Physics, is 7/12.

Notice that this answer (7/12 ≈ 0.583) is quite different from the unconditional probability of a student studying Mathematics (18/30 = 0.6). The condition changed the probability, which is exactly the point of conditional probability. If both answers were the same, the two events would be independent of each other.

Exam Tips for O-Level Probability Questions

Consistent exam performance in probability comes down to disciplined habits rather than raw ability. Students who train themselves to follow a clear process — even under time pressure — rarely make careless errors. Here are the strategies that make the biggest difference:

• Identify the event type first. Always ask: are events independent (with replacement) or dependent (without replacement)? The answer determines whether second-level branch probabilities change.
• Choose your tool before you draw. Use a possibility table when both events produce many outcomes and you need to list sums, products, or differences. Use a tree diagram when events have fewer outcomes or when dependent probability needs to be shown visually.
• Always check branch probabilities sum to 1. This takes five seconds and will catch arithmetic errors before they cascade.
• Label every branch clearly with both the outcome and its probability. Unlabelled diagrams are a source of avoidable mistakes.
• For “given that” questions, underline the condition in the question and write down what the restricted sample space is before applying any formula.
• Verify your final answer makes sense. A probability must be between 0 and 1. If you get a negative value or a value above 1, something has gone wrong.

One more practical tip: in the O-Level exam, probability questions often build across multiple parts (a), (b), (c), with earlier parts setting up the diagram used in later parts. Always complete the full diagram at the beginning, even if only part (a) asks for it — this saves time and reduces the chance of inconsistency in your answers. For more structured practice and guidance on secondary-level topics like this, explore our Secondary Tuition programmes at EduFirst.

Common Mistakes to Avoid

Even well-prepared students can lose marks on probability through a handful of recurring errors. Being aware of these pitfalls puts you in a position to avoid them entirely.

• Using the same probabilities for both draws in a “without replacement” question. This is the single most common error. Always re-examine the new total and the new count of each outcome after the first event in a dependent scenario.
• Multiplying when you should be adding (and vice versa). Remember: AND means multiply (along a path); OR between two separate paths means add.
• Forgetting to include both paths for symmetrical outcomes. “Exactly one red” can happen as Red-Blue or as Blue-Red. Both paths must be included.
• Misreading a conditional probability question as a regular combined probability question. The phrase “given that” narrows the sample space — always account for this.
• Not simplifying fractions in the final answer. O-Level mark schemes often require fractions in their simplest form. Do not leave answers like 6/20 when 3/10 is expected.

Bringing It All Together

Probability in O-Level Maths rewards students who are methodical and deliberate. The concepts themselves — independent and dependent events, possibility tables, tree diagrams, and conditional probability — are all learnable and logical. The real skill is in choosing the right tool for the right question, setting it up carefully, and applying the multiply-or-add rule correctly. With consistent practice using worked examples, most students find that probability becomes one of the more manageable — and even enjoyable — topics in the syllabus.

If you are looking to strengthen your foundation across all O-Level Mathematics topics, EduFirst’s Secondary Tuition programme provides personalised, small-class instruction to help every student build genuine understanding, not just exam technique. And if you have younger siblings at the primary level who could benefit from early mastery of mathematical reasoning, our Primary Tuition programme is built on the same ethos of individualised learning. For students who prefer to study at their own pace, our E-Lessons offer flexible online options with quality content and guidance.