Probability, Year 9: frequency trees and choosing a method

A frequency tree

HintIt grows branches, and each branch carries a count of people or results.

The 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.

110; 35

HintTake the Year 7 figures away from the matching totals, one fork at a time.

The 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.

45 + 25 = 70, not 60

HintEveryone on the parent branch has to go down exactly one of the two forks.

The 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.

2/5

HintWalkers appear at the end of more than one branch — collect them all before dividing.

The 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.

180

HintFind the size of the group you are told about, then see who is left over.

The 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.

The two outcomes are not equally likely

HintLook at the pin's lopsided shape, and ask whether each landing has the same chance.

The 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.

Theoretical probability

HintIt comes from reasoning about a fair situation on paper.

The 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.

Using data that already exists — past weather records

HintYou cannot run the same day again, and wet and dry are not evenly matched.

The 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.

Too few trials

HintAsk how much ten spins can really tell you beside a thousand.

The 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.

estimate

HintIt is an informed approximation, not the exact value.

The whyExperimental results vary from one run to the next, so the probability they give is never exact. More trials make it more reliable.