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This page sets out the method for frequency trees in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Frequency trees topic page.
When you see a partly filled frequency tree with a paragraph of information, work box by box.
When asked for a probability, decide which group the person is chosen from first.
Write the whole group at the start.
Every later branch is a piece of that group.
If 30 people arrive and 18 say yes, 12 say no.
18 + 12 = 30. If it does not, a branch is wrong.
The yes group can split again, for example by age.
Those new branches add to the yes total, not to the original total.
Subtract the known branch from the number that entered.
Do not subtract it from a different group.
“Yes and then under 15” is the end of one path.
Add the ends of two paths if the question says “or”.
These branches are counts, not probabilities.
Do not multiply the frequencies along a path.
A frequency tree is built from counts you can add. If 40 students are surveyed, 40 goes at the start. It is not 40/40 and it is not 1. Writing probabilities too early hides the missing number, because probabilities have already been divided. Stay in whole people until the counts are finished.
Each branch is labelled with the outcome and the count. "Walk, 18" means 18 people walked. It does not mean the probability is 18. The label matters because two trees can share numbers and still answer different questions. Read the words on the branch you are filling in.
Method: 1. Put the grand total at the start of the tree. 2. Write counts on the branches, not fractions. 3. Label each branch with its outcome. 4. Only divide when the question asks for a probability.
Whatever arrives at a point has to leave along the branches. If the outcomes are "yes" and "no", they cover everyone who arrived, so yes plus no equals the arrival. This is why a missing branch is a subtraction. You are not guessing. You are using the fact that the split is complete.
The rule is local. It applies at every point, not only at the start. Thirty-five sandwich buyers split into drink and no drink. Those two branches add to 35, not to the 80 people who entered the shop. Subtracting from the wrong node is an easy mistake, and it produces a number that looks tidy and is still wrong.
Method: 1. Find the point where the missing branch starts. 2. Read the number that arrived at that point. 3. Subtract the known branch from that arrival. 4. Check: known branch plus missing branch equals the arrival.
Fill the tree from left to right. The first missing branch is usually the partner of a number you were given at the start. After that, each end branch is a subtraction inside its own group. Write the number on the diagram as soon as you find it, so the next subtraction uses the right arrival.
You can check a completed end by a second route. The people who do not buy a drink are "sandwich and no drink" plus "no sandwich and no drink". They are also the grand total minus everyone who did buy a drink. If both routes give the same integer, the tree is consistent. If they disagree, one arrival was wrong.
Method: 1. Work from the start towards the ends. 2. At each split, subtract the known count from the arrival. 3. Write each new count on the branch before you move on. 4. Check the answer by a second route that uses different branches.
When the counts are finished, a probability is one count divided by the right total. "What fraction of the people buy tea with sugar?" uses the people who started, because "of the people" means the whole group. Twelve out of fifty is 12/50, which simplifies to 6/25.
The denominator changes if the words change. "Of the tea drinkers" would use the tea branch as the denominator, not the start. Read that phrase before you divide. Simplify the fraction by dividing the top and bottom by the same number. The simplified fraction and the original count must still describe the same share.
Method: 1. Finish the counts before you divide. 2. Choose the denominator from the words: the start, or a named group. 3. Write the end count over that denominator. 4. Simplify, and check the simplified fraction still matches the count.
A frequency tree starts with a total on the left and splits into groups, then splits again. Every pair of branches must add to the number they came from. In an exam you are given clues in sentences, not boxes. Start with the total, then read each sentence and write the number it gives you in the right place. Fill what you can, then use the add-to rule to fill the rest.
Never leave a box empty if it can be found by subtraction. If 80 children split into 35 boys and the rest girls, the girls box is 80 - 35 = 45. If 22 of the boys own a bike, then 35 - 22 = 13 do not. Work left to right, one column at a time, and tick each clue off as you use it. A final check is that the last column adds to the original total.
Method: 1. Write the total in the first box. 2. Use each clue to fill one box, and tick the clue off. 3. Fill any box that can be found by subtracting from the number it came from. 4. Add the last column to check it matches the total.
Tip: When two branches come from the same box, they add to that box. Use this to find a missing branch.
Many questions give a branch as a percentage or a ratio. "25% of the home fans wear a scarf" means 25% of the home fans, not 25% of all the fans. Take the percentage of the number in the box the branch comes from. Home fans number 140, so 25% of them is 0.25 × 140 = 35. Using 200 by mistake gives 50, which loses all the following marks.
A phrase such as "three times as many in Screen 2 as in Screen 1" gives a ratio of 1 : 3. Add the parts, 1 + 3 = 4, then divide the total by 4 to find one part. With 120 customers, one part is 30, so Screen 1 has 30 and Screen 2 has 90. Check that the branches add to 120. Use the original total for ratios, and the box a branch comes from for percentages.
Method: 1. Find which box the percentage or ratio refers to. 2. Convert the percentage to a multiplier, or add the ratio parts. 3. Work out the number and write it on the branch. 4. Subtract to find the other branch and check that the pair adds up.
Sometimes a clue gives a total across the tree, such as "98 of all the people cycle" without saying how many are on each branch. Work out the branches you can find first, add the ones that match the clue, and subtract from the overall total. If 63 of the car owners cycle, then 98 - 63 = 35 of the people who do not own a car cycle.
This takes the same structure as a missing-bar question. Write the known part of the total, subtract it from the clue total, and put the answer in the missing box. Then complete the box next to it by subtraction. A common error is to use the clue total on one branch alone, so check that the numbers on the correct two branches add to the clue.
Method: 1. Fill all the boxes that the clues give directly. 2. Find which boxes the overall total refers to and add the known ones. 3. Subtract from the overall total to find the unknown box. 4. Complete the paired branch by subtraction and check the final column.
Once the tree is complete, probability questions read straight from it. The probability of an outcome is the number in its box over the total that the question chooses from. For a random student from all 120, use 120 as the denominator. If the question says "a student is chosen from those in the choir", the denominator is the number in the choir, not 120.
Read the wording carefully. "From those in the choir, the probability they are in Year 10" is 36/54 = 2/3. Giving 36/120 answers a different question. Give fractions in their simplest form unless asked otherwise, and write the working fraction first so the method mark is safe.
Method: 1. Decide which group the person is chosen from. That number is the denominator. 2. Find the number of people who fit the condition. That is the numerator. 3. Write the fraction, then simplify it. 4. Check that the answer is between 0 and 1.
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