Every card in Probability, Year 9: frequency trees and choosing a method
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- A diagram that starts with a total and splits it into groups at each fork, showing the actual number in every group
A frequency tree
HintIt grows branches, and each branch carries a count of people or results.
WhyIt is a way of keeping track of counts through two questions in a row, such as "which year group?" and then "walks or not?". The numbers at the ends of the branches add back to the starting total.
- 200 pupils were asked whether they walk to school. 90 are in Year 7 and the rest are in Year 8. 50 of the Year 7 pupils walk, and 85 pupils walk altogether. On a frequency tree, the Year 8 branch shows ____ pupils, and the Year 8 'walks' branch shows ____.
110; 35
HintTake the Year 7 figures away from the matching totals, one fork at a time.
WhyThe first fork splits 200 into 90 and 200 − 90. Of the 85 walkers, 50 are in Year 7, so the remaining 85 − 50 must sit on the Year 8 branch. Each number on a tree comes from a total and the part already known.
- On a frequency tree, a branch labelled 60 splits into two branches labelled 45 and 25. How can you tell that a mistake has been made?
45 + 25 = 70, not 60
HintEveryone on the parent branch has to go down exactly one of the two forks.
WhyA fork shares out the people who reach it, with nobody lost and nobody added. So the branches leaving a fork must total the number arriving at it, and that is the check to make at every split.
- A frequency tree for 80 pupils first splits them into 50 who have a pet and 30 who do not. Of the 50 with a pet, 20 walk to school; of the 30 without a pet, 12 walk to school. One of the 80 pupils is picked at random. What is the probability that the pupil walks to school? Give a fraction in its simplest form.
2/5
HintWalkers appear at the end of more than one branch — collect them all before dividing.
WhyThere are 20 + 12 = 32 walkers among the 80 pupils, and 32/80 simplifies to 2/5. A probability for the whole group uses the starting total as its denominator.
- A frequency tree starts with 240 people. The first fork splits them into children and adults, and one quarter of the people are children. What number goes on the "adults" branch?
180
HintFind the size of the group you are told about, then see who is left over.
WhyA quarter of 240 is 60, so 60 children go on one branch and the other 240 − 60 people go on the other. A worded description often gives a fraction or a percentage that has to be turned into a count first.
- A drawing pin can land point up or point down. Why is it wrong to say that the probability of point up must be 1/2?
The two outcomes are not equally likely
HintLook at the pin's lopsided shape, and ask whether each landing has the same chance.
WhyCounting gives the right answer only when every result has the same chance, as with a fair coin. A drawing pin is not symmetrical, so the only way to find its probability is to drop it many times and record what happens.
- A probability worked out by counting equally likely outcomes, without carrying out any experiment
Theoretical probability
HintIt comes from reasoning about a fair situation on paper.
WhyThe chance of a 4 on a fair dice is 1/6 by this method: one winning face out of six that are all as likely as each other. An experiment gives an estimate that settles towards this value as the number of trials grows.
- You want the probability that it rains in Manchester on a day in June. Which is the sensible way to find it: counting equally likely outcomes, running an experiment, or using data that already exists?
Using data that already exists — past weather records
HintYou cannot run the same day again, and wet and dry are not evenly matched.
WhyWet and dry are not equally likely, so counting fails, and a June day cannot be repeated like a dice roll. The proportion of June days that were wet over many years gives a usable estimate.
- A spinner is spun only 10 times and lands on red 6 times. Why is 6/10 a poor estimate of the probability of red?
Too few trials
HintAsk how much ten spins can really tell you beside a thousand.
WhyIn a short run, chance alone can push the results a long way from the true value. The weakness of the experimental method is that it needs a large number of trials before its estimate can be trusted.
- A probability found from an experiment is only an ____: repeat the experiment and the result may come out slightly different.
estimate
HintIt is an informed approximation, not the exact value.
WhyExperimental results vary from one run to the next, so the probability they give is never exact. More trials make it more reliable.
1★ KS3-MATH-PRB-0025
2★ KS3-MATH-PRB-0026
3★ KS3-MATH-PRB-0027
4★ KS3-MATH-PRB-0028
5★ KS3-MATH-PRB-0029
6★ KS3-MATH-PRB-0030
7★ KS3-MATH-PRB-0031
8★ KS3-MATH-PRB-0032
9★ KS3-MATH-PRB-0033
10★ KS3-MATH-PRB-0034
Keep what you learn
Here, nothing is saved. In your child’s own sky every card is scheduled — it comes back just before they’d forget it — and the professor who wrote it is one tap away.