GCSE · Maths · AQA · Spec 8300

Frequency of outcomes; tables and frequency trees

Sixty customers, two questions each, and one quick check that tells you straight away if anybody has gone missing.

Frequency tree

Follow every customer down the tree

A café counts its first 60 customers. 36 buy a hot drink and the rest buy a cold drink. 15 of the hot-drink customers also buy cake, and 8 of the cold-drink customers buy cake. Work out each missing number before you tap its branch, then tap to check.

Drink → Cake?

Each customer goes down exactly one branch at every fork, so nobody is lost and nobody is counted twice.

4 ends, and every customer finishes on exactly one of them.

On All 60 customers. 2 branches to choose from.

The rule that holds the whole tree together: at every fork, the branches add back up to the number they came from.

Relationship matrix

Tap any cell to reveal it. Tap a column header to read one property down every item.

	What the frequency tells you	Out of all 50 spins
Redfrequency 18
Bluefrequency 12
Greenfrequency 9
Yellowfrequency 11
Totalevery spin, added up

Each cell hides a short answer and the reason behind it. Predict before you tap.

Back to the café

What goes on the bottom?

The café tree again: 60 customers at the top, and 15 of them on the 'hot drink and cake' end. One of the 60 customers is picked at random.

What is the probability that they had a hot drink and cake? Pick the idea closest to what you think right now.

15/36, because the cake end comes off the hot-drink branch, so it's out of 36.15, because that's how many had a hot drink and cake.15/60, which simplifies to 1/4, or 0.25, or 25%.15 : 45, because 15 had both and 45 didn't.15 out of 60. I'd just write it in words.

How sure are you?

I'm sureI think soI'm guessing

15/36, because the cake end comes off the hot-drink branch, so it's out of 36.

This is a common misconception

Tempting, because on the tree the 15 really does hang off the 36. But the customer wasn't picked from the hot-drink customers only. They were picked from all 60. The bottom of the fraction is whoever you are picking FROM, and here that is everybody: 15/60.

15, because that's how many had a hot drink and cake.

This is a common misconception

15 is the count on that end, and you've read it perfectly. But a count on its own doesn't say 'out of how many'. 15 out of 60 is very different from 15 out of 1000, so a probability needs the total underneath: 15/60.

15/60, which simplifies to 1/4, or 0.25, or 25%.

This matches the evidence

Yes. The customer is picked from all 60, so 60 goes on the bottom and the 15 on that end goes on top. Divide top and bottom by 15 to get 1/4. For the decimal, do top ÷ bottom: 1 ÷ 4 = 0.25. For the percentage, multiply by 100: 25%. All four are the same probability, just written differently.

15 : 45, because 15 had both and 45 didn't.

This is a common misconception

Your numbers are right: 15 did, and 60 − 15 = 45 didn't. But a ratio compares two groups with each other. A probability compares one group with the WHOLE group, so it is 15 over 60, not 15 to 45. Write it as a fraction, a decimal or a percentage.

15 out of 60. I'd just write it in words.

This is a common misconception

Your thinking is exactly right: 15 out of the whole 60. Only the writing needs to change. A probability is written as a fraction, a decimal or a percentage, so '15 out of 60' becomes 15/60, or 1/4, 0.25 or 25%.

Picking one at random from the whole group? Put its count over the whole-group total, and write it as a fraction, a decimal or a percentage: 15/60 = 1/4 = 0.25 = 25%.

Spot the slip

One line sends the answer wrong

80 students chose a trip: 45 went to the museum and the rest went to the zoo. 27 of the museum group took a packed lunch, and 21 of the zoo group took a packed lunch. Complete the frequency tree, then find the probability that a student picked at random from all 80 went to the zoo and did not take a packed lunch.

Alex's answer — which line goes wrong?

Zoo: 80 − 45 = 35 students
Museum, no packed lunch: 45 − 27 = 18
Zoo, packed lunch: 21 (given in the question)
Zoo, no packed lunch: 80 − 21 = 59
Probability (zoo, no packed lunch) = 59/80
= 0.7375

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Every trial lands in exactly one place, so the counts always add back to where they came from.

What you need to know

A frequency is how many times an outcome happened. A frequency table lists every outcome with its frequency, and the frequencies add up to the total number of trials.
A frequency tree sorts one whole group twice. At every fork, the two branch counts add back to the number they split from.
Every person or trial finishes on exactly one end of the tree, so all the ends together add up to the total at the top.
Picked at random from the whole group? The probability is the count you want over the whole-group total, written as a fraction, a decimal or a percentage, never as a ratio or in words.

The big picture

A frequency table records how many times each outcome happened, and a frequency tree sorts one whole group by one question and then by another. Every trial lands in exactly one place, so the counts always add back up: two branches to the branch they came from, all the ends to the total at the top. To find the probability that one picked at random from the whole group is in a particular place, put its count over the whole-group total, and write it as a fraction, a decimal or a percentage.

Key points

1Frequency = how many times it happened.

2Most frequent outcome = the one with the biggest frequency. Total = every frequency added together.

3Fork check: branch + branch = the number above them.

4Missing count? Subtract from its own branch, never from the total at the top.

5Probability from the whole group = count over the whole-group total, e.g. 15/60 = 1/4 = 0.25 = 25%.

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Worked example

Problem

A teacher records how each of the 40 students in a class got to school: 14 walked, 9 came by bus, 5 cycled and the rest came by car. (a) How many came by car? (b) One student is picked at random from the class. Find the probability that they came by car, as a fraction, a decimal and a percentage.

⚠ Watch out

Splitting from the wrong number. When a branch of 35 splits and one end is 21, the other end is 35 − 21, not the top total minus 21. And when someone is picked from everyone, divide by everyone, not by the branch they sit on.

🧠

Memory hook

The top of the tree goes on the bottom of the fraction, and every fork adds back up to where it came from.

✓

Check yourself

A frequency tree starts with 50 people, and its first two branches say 28 and 24. How can you tell at once that something has gone wrong?

Flashcards

(7)

What does the frequency of an outcome tell you?

How many times that outcome happened in the experiment.

In a frequency table, what do all the frequencies add up to?

The total number of trials, because every trial is counted in exactly one row.

How do you spot the most frequent outcome in a frequency table?

Find the biggest frequency. The outcome in that row is the most frequent one.

What must be true at every fork of a frequency tree?

The two branch counts add back up to the number they split from.

A branch of 40 splits into two ends. One end is 26. How do you find the other?

40 − 26 = 14. Subtract from its own branch, not from the total at the top of the tree.

One person is picked at random from the whole group. What goes on the bottom of the probability fraction?

The whole-group total: the number at the very top of the tree.

Which forms can you write a probability in?

A fraction, a decimal or a percentage, such as 1/4, 0.25 or 25%. Never a ratio like 15 : 45 or words like '15 out of 60'.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Follow every customer down the tree

What goes on the bottom?

One line sends the answer wrong

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