Probability — Edexcel GCSE Mathematics
Test yourself on Probability with PEARSON EDEXCEL GCSE practice questions.
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Probability explained
Frequency data for two categories can be shown in a two-way table or a frequency tree.
Read the full explanation
A two-way table uses rows for one category (e.g., Male/Female) and columns for another (e.g., Pass/Fail). Each cell shows the count for a specific combination. The rows and columns must sum to their respective totals, and the grand total must be consistent. A frequency tree shows the same data differently. Start with the total population on the left. The first branches split this total by one category (e.g., 80 people into 50 Male, 30 Female). The next set of branches splits each of these groups by the second category (e.g., 50 Males into 12 Pass, 38 Fail). In both representations, missing values are found by subtraction, ensuring all parts add up correctly.
apply ideas of randomness, fairness and equally likely events to calculate expected outcomes of multiple future experiments
To calculate the expected number of times an event will occur, use the formula: Expected outcomes = P(event) × number of trials. For example, if the probability of a spinner landing on red is 0.2, in 50 spins the expected number of reds is 0.2 × 50 = 10. This is a theoretical average; the actual number may differ. Key terms must be used precisely. 'Random' means the next outcome is unpredictable. 'Equally likely' means each outcome has the same probability. 'Fair' usually implies equally likely outcomes, as with a fair dice or coin. A spinner with different-sized sectors has outcomes that are not equally likely, but it is 'unbiased' if it behaves according to its theoretical probabilities. A large difference between an observed result and the expected outcome is evidence of bias.
relate relative expected frequencies to theoretical probability, using appropriate language and the 0-1 probability scale
Every probability sits between 0 and 1, where 0 means impossible and 1 means certain, and words such as unlikely, even chance and likely mark places along that line. Theoretical probability comes from counting equally likely outcomes. Relative frequency comes from an experiment, and it is the number of successes divided by the number of trials. The two answer the same question from opposite directions, and you compare them to judge whether an object is fair. A drawing pin lands point up 37 times in 100 drops, so the relative frequency is 0.37, and that is your best estimate of the probability because there is no way to count equally likely outcomes for a pin. Give probabilities as fractions, decimals or percentages, and not as odds or as a ratio.
apply the property that the probabilities of an exhaustive set of outcomes sum to one; apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one
When a list of outcomes covers everything that can happen, their probabilities add to 1, so a missing one is found by subtracting the rest from 1. Two events are mutually exclusive when they cannot both happen at the same time, and for events like that you are allowed to add their probabilities. The probability that an event does not happen is 1 minus the probability that it does. Take a bag of red, blue and green counters. If red has probability 0.2 and blue has probability 0.45, then green has probability 0.35, and the probability of not red is 0.8. Watch for a table where two of the entries are written in terms of the same letter. There you form an equation saying the entries total 1, then solve it and put the values back in the table.
understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size
The more trials you run, the closer the relative frequency from an unbiased experiment settles towards the theoretical probability. That is why a result from 500 throws is a better estimate than one from 20 throws, and why two short experiments can disagree with each other, and with the theory, without either being wrong. Questions give a table of results after different numbers of trials and ask which estimate to trust, or ask you to pool the results by adding all the successes and dividing by all the trials. Ten throws of a coin giving 7 heads is a relative frequency of 0.7, which is far too short a run to show bias, whereas a thousand throws giving 700 heads would be strong evidence of it. The relative frequency settles closer and closer to the theoretical value rather than locking onto it.
enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams
To list outcomes systematically, fix the first choice and cycle through all second choices before moving to the next first choice. A two-way table does this with rows and columns. A tree diagram is for sequential events: the first branches show the first choice, and from each of these, new branches show the second. For example, for a coin toss and a dice roll, two initial branches for H/T would each lead to six further branches for 1-6. Each path from start to finish is a unique outcome. A Venn diagram sorts items by properties. Always fill the intersection (overlap) first, then subtract this from each circle's total to find the 'only' regions. Anything left over goes outside. The union (∪) is everything in either set; the intersection (∩) is only the overlap.
construct theoretical possibility spaces for single and combined experiments with equally likely outcomes and use these to calculate theoretical probabilities
A possibility space lists every equally likely outcome. For a single experiment, such as rolling one fair six-sided die, the space is simply 1, 2, 3, 4, 5, 6, so the probability of an even number is 3/6 = 1/2. For two combined experiments, draw a grid: put one die down the side and the other along the top, then write the total in each of the 36 cells. Every cell is equally likely, so a probability is the number of cells you want divided by 36. Four cells hold a total of nine, so P(total 9) = 4/36 = 1/9. This method depends on the cells being equally likely. The totals themselves are not equally likely, which is exactly why you count cells rather than count the different totals available.
calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions
For sequential events, multiply probabilities along each tree path, then add non-overlapping paths when any satisfies the question. The general rule is P(A and B) = P(A) × P(B given A). If A and B are independent, knowing A happened does not change B. For two independent flips of a biased coin with P(Head) = 0.6, the second-flip branches repeat the first and P(two Heads) = 0.6 × 0.6 = 0.36. For dependent events, branch probabilities change. From a bag with 5 red and 3 blue counters, without replacement, P(Red first) = 5/8 and P(Red second given Red first) = 4/7, so P(two Reds) = 5/8 × 4/7 = 20/56. A sample-space diagram or two-way table can be clearer for simultaneous events such as two dice. Treat its outcomes as equally likely only when the dice are fair. State the assumptions used: random selection, replacement, equally likely outcomes and justified independence.
calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams (Higher tier only)
Conditional probability, P(A | B), is the probability of event A happening, given that B has already occurred. This restricts the sample space to outcomes where B is true. Problems can involve observed counts or expected frequencies, which are theoretical counts calculated from probabilities (e.g., if P(X) = 0.1, the expected frequency in 50 trials is 0.1 × 50 = 5). From a two-way table, P(A | B) is the count in cell (A and B) divided by the total for condition B. From a Venn diagram, use P(A | B) = P(A ∩ B) / P(B). On a tree diagram, P(B | A) is read from the second branch. For an 'inverse' probability like P(A | B), use the formula: find P(A and B) by multiplying along branches, then divide by P(B), which may require summing multiple path probabilities.
Your focus
- Describe how a frequency tree splits a total in two stages and carries counts rather than probabilities.
- Complete a frequency tree from the wording of a question and read a required frequency off the finished tree.
- Explain why each pair of branches must add back to the number it came from, using that to find a missing value.
Show all 32 objectives
- Distinguish between randomness, fairness, bias and equally likely events.
- Calculate an expected number of outcomes by multiplying the event's probability by the number of trials.
- Explain that an expected outcome is a long-run average and why a single experimental result may differ.
- Describe the 0 to 1 probability scale, placing impossible, unlikely, even chance, likely and certain along it.
- Estimate a probability as relative frequency, successes over trials, where equally likely outcomes cannot be counted.
- Compare that estimate with the theoretical probability and conclude in words whether the object looks biased.
- Describe what mutually exclusive means and state that an exhaustive set of probabilities totals one.
- Calculate a missing probability by subtracting the listed ones from 1, written in the same form as the others.
- Work out an unknown entry given in terms of a letter by forming an equation totalling 1 and solving it.
- Explain why the probability that an event does not happen is 1 minus the probability that it does.
- Describe how relative frequency settles closer to the theoretical probability as the number of trials grows.
- Estimate a probability by pooling results as total successes over total trials rather than averaging the fractions.
- Justify which of several estimates to trust by pointing to the one taken from the largest number of trials.
- Describe a systematic order for listing outcomes, holding the first item while the last one varies.
- Complete a two-way table or grid by labelling rows and columns and filling every cell with an outcome.
- Construct a tree diagram for two or more events and use it to list all possible outcomes.
- Complete a Venn diagram by filling the overlap first and subtracting it from each set total.
- Explain the difference between the union and the intersection of two sets when reading a Venn diagram.
- Describe a possibility space as the full list of equally likely outcomes, for one experiment or set out as a grid for two.
- Calculate a probability from a possibility space as the cells you want over the total number of cells.
- Explain why the cells are counted rather than the different totals, since the totals are not equally likely.
- Distinguish between independent and dependent events and adjust calculations accordingly.
- Construct a tree diagram for a sequence of events, calculating probabilities on each branch.
- Use a tree diagram or a sample space diagram to calculate the probability of combined events.
- State and test the assumptions used in a probability model, including random selection, equally likely outcomes, replacement and independence.
- Calculate expected frequencies from given probabilities and totals, and use them in two-way tables.
- Calculate a conditional probability from a two-way table, Venn diagram or tree diagram.
- Distinguish between finding P(B|A) and P(A|B) on a tree diagram.
- Use the formula P(A | B) = P(A and B) / P(B) to solve inverse probability problems.
Probability exam tips
Quick Revision Summary (Key Takeaway)
Probability is the branch of mathematics that quantifies how likely events are to occur, expressed as a number between 0 and 1 on a probability scale. In Pearson Edexcel GCSE Mathematics, you must calculate probabilities using fractions, decimals or percentages, use Venn diagrams and tree diagrams for combined events, and apply the addition and multiplication rules including conditional probability.
Topic Overview
Probability is the study of chance and uncertainty, forming a core part of the Pearson Edexcel GCSE Mathematics curriculum. It teaches you how to assign numerical values to the likelihood of events, from simple single events to complex combined and conditional scenarios. You will learn to use probability scales, Venn diagrams, tree diagrams, and two-way tables to model real-world situations.
This topic is essential for developing logical reasoning and problem-solving skills, and it appears frequently in both Foundation and Higher tier exams. It connects to other areas such as statistics, algebra (for setting up equations), and set theory. Mastering probability also provides a foundation for understanding risk and data in everyday life and further study.
Key Concepts
- →Probability is measured on a scale from 0 (impossible) to 1 (certain), and can be written as a fraction, decimal, or percentage.
- →The sum of probabilities of all mutually exclusive outcomes is 1, so P(not A) = 1 - P(A).
- →For independent events, P(A and B) = P(A) * P(B); for dependent events, the probability of the second event changes based on the first.
- →The addition rule: P(A or B) = P(A) + P(B) - P(A and B), which accounts for overlap.
- →Tree diagrams and Venn diagrams are powerful tools for organising and solving multi-stage probability problems.
Marking Points
- Correctly calculating a missing value in a two-way table using row or column totals.
- Placing the correct overall total at the start of a frequency tree.
- Ensuring each pair of branch frequencies on a tree adds back to the number at the node they came from.
- Reading the correct frequency from the completed table or tree to answer a question, such as finding a probability.
- identifying the correct theoretical probability of the single outcome
- showing the correct calculation: probability multiplied by the number of trials
- stating the final expected number of outcomes, which can be a non-integer
- explaining that the expected value is a theoretical average and the actual result may differ
- a probability written in an accepted form, as a fraction, a decimal or a percentage
- the relative frequency as successes over trials, credited whether or not it is simplified
- a comparison that names both values and says which is the larger
- a conclusion in words about fairness or bias that is supported by that comparison
- setting the sum of the listed probabilities to 1, shown either as an equation or as a subtraction
- the correct total of the probabilities that are given, credited even if the subtraction after it is wrong
- the missing probability, written in the same form as the ones already in the table
- solving the resulting equation correctly when the entries are written in terms of a letter
- choosing the estimate that comes from the largest number of trials, with a reason given
- pooling results as total successes over total trials, rather than averaging the separate fractions
- a relative frequency written correctly as a fraction, decimal or percentage
- a comment linking closer agreement with the theoretical value to a larger sample
- a systematic list, credited for an order that shows a clear pattern rather than for the count alone
- a complete set of outcomes with none missing and none repeated
- a two-way table or grid with rows and columns labelled and every cell filled consistently
- a correctly structured tree diagram with branches for each stage and labels for each outcome
- each Venn diagram region filled consistently with the totals given, starting with the intersection
- a value found by subtracting the overlap from a set total, awarded even if a later region is wrong
- a correctly built possibility space for a single experiment, listing every equally likely outcome once
- a correctly built grid for a combined experiment, with every combination appearing once
- identifying the outcomes that satisfy the condition, shown by ringing or listing them
- a probability whose denominator is the total number of equally likely outcomes
- the answer in its simplest form when the question asks for it
- Correctly calculating probabilities for the first event or set of branches.
- For dependent events (without replacement), correctly adjusting probabilities on the second branches: reducing the denominator by one, and reducing the numerator only for the same outcome.
- Multiplying probabilities along a branch to find the probability of a specific combined outcome.
- Adding the probabilities of all relevant paths that satisfy the question's condition (e.g., 'probability of getting one of each colour').
- Correctly constructing and using a sample space diagram or two-way table to find a required probability.
- Calculating an expected frequency from a probability and a number of trials.
- Correctly identifying the restricted group (denominator) created by the condition in a table or Venn diagram.
- On a tree diagram, correctly calculating P(A and B) by multiplying along branches.
- For an inverse probability problem, calculating the total probability for the denominator, e.g. P(B), by summing relevant paths.
- Using the formula P(A | B) = P(A and B) / P(B) correctly to find a conditional probability.
Examiner Tips
- 💡When completing a two-way table, check that the sum of your row totals equals the sum of your column totals. This is a vital consistency check.
- 💡After completing a frequency tree, check that the numbers at the end of all the final branches add up to the starting total.
- 💡Clearly write down the probability and the number of trials before you multiply. This can secure method marks even if your final answer is wrong.
- 💡An expected outcome can be a decimal or fraction (e.g., 7.5 times). This is mathematically correct as it represents an average. Do not round it to a whole number unless the question specifically asks for an interpretation that requires it.
- 💡On a probability scale, work the value out as a decimal first and then measure the position along the printed line rather than guessing it.
- 💡If you are asked which of two estimates is better, choose the one from the larger number of trials and say the number of trials in your reason.
- 💡Add the given probabilities on your calculator before you subtract, then check the completed set adds back to 1.
- 💡Answer in the form the question uses. If the table is in fractions with the same denominator, give a fraction rather than converting to a decimal.
- 💡When asked which estimate is best, quote the number of trials in your reason. Saying it is bigger, with no number, is not a reason.
- 💡If a table adds trials in stages, keep a running total column as you go, so the pooled figures are already written down when a later part needs them.
- 💡Count your list against the number you get from multiplying the choices together. If the two disagree, check your systematic method.
- 💡Build a Venn diagram from the middle outwards, and check all regions add to the grand total before you answer any part.
- 💡When drawing a tree diagram, write the combined outcome at the end of each final branch to help you list them all systematically.
- 💡Draw the grid even when the question does not ask for one. The probability marks come faster off a grid than off a written list.
- 💡Ring the cells you are counting rather than counting them in your head, so you can recount them if the number looks wrong.
- 💡Before starting, find the words 'with replacement' or 'without replacement'. This is the key to deciding if the probabilities on the second set of branches change or stay the same.
- 💡For questions asking for the probability of 'at least one' of something, it is almost always quicker to calculate 1 – P(none of them) instead.
- 💡Choose the best representation. Tree diagrams are excellent for sequential events. Sample space diagrams (or two-way tables) are often clearer for simultaneous events like rolling two dice.
- 💡Underline the words that follow 'given that' or the vertical bar '|'. Whatever they describe defines your denominator.
- 💡For inverse probability on a tree diagram, draw out all the paths that lead to the 'given' event to make sure you sum their probabilities correctly for the denominator.
- 💡Always read the question carefully to determine whether events are independent or dependent. Look for keywords like 'without replacement' or 'given that'.
- 💡Show all working, especially when using tree diagrams. Write probabilities on each branch and clearly identify which branches you are multiplying or adding.
- 💡Simplify fractions in your final answer unless the question asks for a decimal or percentage. Check that your answer is between 0 and 1.
Common Mistakes
- Writing probabilities (e.g., 0.4) or percentages on the branches of a frequency tree instead of frequencies (counts). Correction: Frequency trees always show counts of items or people.
- Making an arithmetic error when calculating a missing value by subtraction in a table or tree. Correction: Double-check your subtractions, especially under exam pressure.
- Answering with a frequency when the question asked for a probability, or vice versa. Correction: If the question asks for 'how many', give the count. If it asks for 'the probability', give a fraction (e.g., 19/80).
- dividing the number of trials by the probability instead of multiplying
- incorrectly assuming outcomes are equally likely (e.g., for a spinner with different-sized sectors or a biased dice)
- confusing 'random' with 'equally likely'; a process can be random without all outcomes having the same chance
- writing a probability as 1 in 4, or as 3 to 1, which are ratios and not probabilities
- giving a probability above 1, usually by dividing the number of trials by the number of successes
- placing an arrow on a probability scale by eye, so a value such as 0.2 ends up nearer the middle than it should be
- saying an object is biased without quoting the theoretical probability it is being compared with
- adding the probabilities of two events that can happen together, such as scoring an even number and scoring above 3 on a dice
- subtracting from 100 when the probabilities are decimals, or subtracting from 1 when they are percentages
- assuming the outcomes in a table are all there are, when the question names another possibility that is not listed
- reading not red as blue when the bag holds three colours, so subtracting from 1 gives the wrong outcome's probability
- averaging the relative frequencies of each stage instead of using the combined totals
- claiming a small sample proves bias because its result is far from the expected one
- expecting the running relative frequency to land exactly on the theoretical probability and stay there
- choosing the estimate closest to the theoretical value rather than the one from the most trials
- listing outcomes in no particular order, which often leads to omissions or repetitions
- putting a whole set total in the part of the circle outside the overlap, so the overlap is counted twice
- in a tree diagram, forgetting to draw a set of branches for the second event from *every* outcome of the first event
- forgetting the region outside both circles in a Venn diagram, which holds the items with neither property
- using the number of different totals as the denominator, for example 11 rather than 36 for two dice
- counting a pair such as one and three only once when the two dice are distinguishable, so the count comes out short
- adding two probabilities when the question needs both events to happen together
- writing the answer as a ratio of wanted cells to unwanted cells instead of as a probability
- In 'without replacement' problems, keeping the denominator the same for the second event. Correction: The total number of items has decreased by one, so the denominator must also decrease by one for the second event.
- Adding probabilities along the branches of a tree diagram instead of multiplying. Correction: The 'AND' rule for a sequence of events means you must multiply along the path. The 'OR' rule means you add the probabilities of separate, successful paths.
- Forgetting to consider all valid outcomes, e.g., for P(one of each colour), calculating P(Red then Blue) but not P(Blue then Red). Correction: If the order is not specified, you must find the probability of all possible orderings that satisfy the condition and add them together.
- Dividing by the overall total instead of the total for the conditioned group. Correction: The 'given that' part of the question defines the new total (the denominator).
- Confusing P(A | B) with P(B | A), especially on tree diagrams. Correction: P(B | A) is read from the second branch following A, while P(A | B) requires the formula.
- Calculating P(A and B) but forgetting to divide by P(B) for an inverse probability question. Correction: Remember the full formula P(A | B) = P(A and B) / P(B).
- Students often think that if an event has not happened for a while, it becomes more likely (gambler's fallacy). In reality, independent events remain unaffected by previous outcomes.
- When using the 'or' rule, students frequently forget to subtract the intersection, leading to probabilities greater than 1. Always check for overlap.
- For dependent events, students may incorrectly keep the denominator the same on tree diagrams. Remember that without replacement, both the numerator and denominator change.
Revision Plan
- 1Day 1-2: Revise basic probability concepts: probability scale, mutually exclusive events, and the complement rule. Practice simple single-event problems.
- 2Day 3-4: Master Venn diagrams and two-way tables. Practice finding probabilities from these representations, including conditional probabilities.
- 3Day 5-6: Learn tree diagrams for independent and dependent events. Practice drawing them and calculating combined probabilities.
- 4Day 7-8: Study the addition rule and conditional probability formula. Solve problems involving 'or' and 'given that'.
- 5Day 9-10: Complete past paper questions on probability, focusing on exam technique and timing. Review mistakes and revisit weak areas.
Exam Question Types
- 📋Simple probability calculations from a table, spinner, or bag of counters. Advice: Identify total outcomes and favourable outcomes, and simplify fractions.
- 📋Tree diagram problems for two or more events, often without replacement. Advice: Draw the tree neatly, label branches with fractions, and multiply along branches for 'and' questions.
- 📋Venn diagram questions involving set notation and conditional probability. Advice: Fill in the Venn diagram systematically, starting with the intersection, and use the formula P(A|B) = P(A and B) / P(B).
- 📋Probability equations, such as finding an unknown number of items given a probability. Advice: Set up an equation using the probability definition and solve for the unknown.
Command Word Expectations (PEARSON EDEXCEL)
You must work out a numerical answer, showing sufficient working. For probability, this often means using a formula or tree diagram and simplifying the final fraction.
You must demonstrate the steps to arrive at a given result, often by setting up an equation or using a probability rule. All steps must be clear and logically correct.
This indicates conditional probability. You must restrict your sample space to the condition and calculate the probability accordingly, often using the formula P(A|B) = P(A and B) / P(B).
How Students Lose Marks (Examiner Pitfalls)
Step-by-Step Worked Solutions
Question: A bag contains 5 red counters and 3 blue counters. Two counters are taken at random without replacement. Calculate the probability that both counters are red.
- 1.Step 1: Identify that this is a dependent event because the first counter is not replaced, so the total number of counters decreases.
- 2.Step 2: Calculate the probability of the first counter being red: P(Red first) = 5/8.
- 3.Step 3: Calculate the probability of the second counter being red given the first was red: P(Red second | Red first) = 4/7 (since 4 red remain out of 7 total).
- 4.Step 4: Multiply the probabilities along the branch: P(Red and Red) = (5/8) * (4/7) = 20/56 = 5/14.
Question: In a class of 30 students, 18 study French, 12 study Spanish, and 6 study both French and Spanish. A student is chosen at random. Find the probability that the student studies French or Spanish (or both).
- 1.Step 1: Identify the given probabilities: P(F) = 18/30, P(S) = 12/30, P(F and S) = 6/30.
- 2.Step 2: Apply the addition rule for two events: P(F or S) = P(F) + P(S) - P(F and S).
- 3.Step 3: Substitute the values: P(F or S) = 18/30 + 12/30 - 6/30 = 24/30.
- 4.Step 4: Simplify the fraction: 24/30 = 4/5.