Here are some fun problems which could start you off on teaching tree diagrams:
1) You wake up in the dark at 6.30am to get ready for school and find that the electricity isn´t working. You have to get changed in the dark and when it comes to picking some socks to wear you just stick your hand into the drawer and pick one sock after the other. You have 14 red socks and 6 yellow socks. What is the probability that you will pick out two socks of the same colour? (One of my year 9´s asked why they only have red and yellow socks – I replied by saying that they must be a member of Gryffindor House).
2) You want to go to a party on Friday night but your parents want you to stay in and study for a test on Monday. As your Mum is a maths teacher, she devises a game to decide whether you will go out or not. She gives you a pack of 15 cards, 6 are red and 9 are black. She says that you can go out if you pick a red and black in any order. Is it more likely that you’ll be staying in or going out?
3) Three people are asked to pick a number from 1 to 10. What is the probability that they all pick the number 4?
I usually teach tree diagrams by thouroughly going through the first question and then getting the students to figure out the second one.
I love question three for a number of reasons. Firstly, it’s a fine example of not having to draw out an entire tree diagram to solve a problem. Secondly, in my experience over the last 3 years of using this question, every student has made assumptions. For one the students assume that the numbers have to be whole numbers. Of course there is no reference in the question to whole numbers. Indeed, loads of clever-clogs pick numbers like pi (3.14159…). Secondly, assuming that you can only pick whole numbers, the students then assume that each whole number from one to ten is equally likely. In real-life, it often turns out that the number 7 is most frequently picked and the numbers 1 and 10 are least frequenctly picked.
Here´s an example of an experiment carried out with 1000 Turkish people randomly picking a whole number between 1 and 10.
It´s interesting that the more people who know about this, the less the number seven will be picked. If we spread the word enough it´s likely that less and less people will pick seven and then it will be an under-picked number!
Does anyone have anymore interesting questions to use as examples when teaching tree diagrams?
One more thing to add, I recently saw a nice task on the nrich website on tree diagrams which I set as a homework today – Chances Are.