Probability, Year 9: frequency trees and choosing a method

A frequency tree

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

为什么It 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

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

为什么The 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

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

为什么A 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

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

为什么There 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

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

为什么A 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

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

为什么Counting 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

提示It comes from reasoning about a fair situation on paper.

为什么The 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

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

为什么Wet 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

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

为什么In 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

提示It is an informed approximation, not the exact value.

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