Probability Trees

Understanding probability trees

Probability trees are an important statistics topic in GCSE and IGCSE mathematics. They help students organise questions where more than one event happens. A probability tree shows the possible outcomes step by step, with branches for each result and probabilities written on the branches. This makes it easier to calculate combined probabilities without losing track of the different routes. Students often meet probability trees after learning basic probability, fractions, decimals and percentages. The topic becomes more demanding because the student must decide whether events are independent or dependent. Independent events do not affect each other. For example, flipping a fair coin twice has the same probability each time. Dependent events do affect each other. For example, taking counters from a bag without replacement changes the numbers left in the bag, so the second probability changes. Probability trees are useful because they turn word problems into a clear visual structure. Instead of trying to hold every possibility in memory, the student can draw branches, label them carefully and follow each route from left to right. This is especially helpful in exam questions involving counters, balls, cards, spinners, weather, tests, choices or repeated trials.

Key rules and formulas

Probability of one event: favourable outcomes divided by total outcomes. Along a branch route: multiply probabilities. This finds the probability of a sequence of events happening together. Different successful routes: add the route probabilities. This finds the probability of one route or another route happening. All probabilities from a single set of branches should add to 1. This is a useful check when completing a tree diagram. For independent events, the probabilities stay the same on the next stage. For dependent events, the probabilities change because the outcome of the first event affects what is left or what happens next. A strong method is to draw the tree first, label every branch, multiply along each route and then add only the routes that match the question. This helps students avoid guessing and makes the working clear for method marks.

Worked examples

Example 1: A fair coin is flipped twice. Find the probability of getting two heads. The probability of heads on the first flip is 1/2. The probability of heads on the second flip is also 1/2 because the events are independent. Multiply along the route: 1/2 multiplied by 1/2 equals 1/4. The probability of two heads is 1/4. Example 2: A bag contains 3 red counters and 2 blue counters. One counter is chosen, replaced, and then another counter is chosen. Find the probability of two red counters. Because the first counter is replaced, the probabilities stay the same. The probability of red is 3/5 each time. Multiply 3/5 by 3/5 to get 9/25. Example 3: A bag contains 4 green counters and 6 yellow counters. One counter is chosen and not replaced, then a second counter is chosen. Find the probability of two green counters. The first probability is 4/10. After one green counter is removed, there are 3 green counters left out of 9 counters. The second probability is 3/9. Multiply 4/10 by 3/9 to get 12/90, which simplifies to 2/15.

Practice exercises and answers

Exercise 1: A fair coin is flipped twice. Find the probability of getting one head and one tail in any order. Answer 1: There are two successful routes: head then tail, and tail then head. Each route has probability 1/2 multiplied by 1/2 equals 1/4. Add the two routes: 1/4 plus 1/4 equals 1/2. Exercise 2: A spinner has probability 0.3 of landing on red and 0.7 of landing on blue. It is spun twice. Find the probability of red then blue. Answer 2: Multiply along the route. 0.3 multiplied by 0.7 equals 0.21. Exercise 3: A bag contains 5 black counters and 3 white counters. Two counters are chosen without replacement. Find the probability of two white counters. Answer 3: First white is 3/8. After one white is removed, the second white is 2/7. Multiply 3/8 by 2/7 to get 6/56, which simplifies to 3/28.

Where this topic appears in school maths

Probability trees are usually taught during GCSE and IGCSE statistics. They may appear in foundation or higher papers, although higher-level questions often include more complex dependent events or algebraic probabilities. Students may be asked to complete a tree diagram, use a tree diagram to calculate a probability or draw the tree themselves from a written question. This topic links strongly with fractions because many probabilities are written as fractions. Students need to multiply fractions, simplify answers and sometimes convert between fractions, decimals and percentages. It also connects with ratio, sample spaces, Venn diagrams and conditional probability. Exam questions often test whether the student understands the wording. Phrases such as with replacement and without replacement are very important. With replacement means the item is put back, so the probabilities usually stay the same. Without replacement means the item is not put back, so the numbers change for the next event. Careful reading is just as important as calculation.

Common mistakes

A common mistake is adding probabilities when they should be multiplied. If the question follows one route through the tree, multiply along that route. Add only when combining separate routes that both satisfy the question. Another mistake is forgetting to change probabilities when there is no replacement. If a counter is removed, both the number of favourable outcomes and the total number may change. Students also sometimes include the wrong routes. For example, if the question asks for exactly one red, the successful routes might be red then not red, and not red then red. It would be wrong to include red then red. Highlighting the required routes on the tree can help prevent this. Rounding too early can also cause inaccurate answers. Students should keep fractions or exact decimals during the working and only round the final answer if the question asks them to do so.

Why one-to-one online lessons help

Online maths lessons can be very beneficial for probability trees because the calculation is only part of the challenge. Students must understand the wording, draw the branches, decide whether the events are independent or dependent, and choose the correct routes. A one-to-one tutor can watch how the student sets out the tree and correct misunderstandings immediately. For future study, probability trees support statistics, decision making, science, economics, data analysis and risk-based reasoning. They help students understand how events combine and how one event can affect another. Regular online tutoring builds confidence with fractions, problem solving, exam wording and statistical thinking. This helps students move from memorising a method to understanding why the tree structure works.