AO elaboration and other teaching resources

AO elaboration and other teaching resources

In this unit, students will explore the commonly held belief that if anything bad can possibly happen it will and at the most inopportune time. They will be encouraged to look at events involving chance and predict the likelihood of certain outcomes by both trialling the event and analysing it theoretically. There is a lot of work in this unit so you might like to spread it over more than one week or do different parts at different times of the year. Some of the simulations are reasonably complicated so it may take students most of a lesson until they have come to grips with the rules involved.

- use simulations to investigate probability in common situations
- predict the likelihood of outcomes on the basis of an experiment
- determine the theoretical probability of an event

This unit takes an interesting approach to probability by delving into some commonly held myths that can be thought of as going under the name of Murphy’s Law. It is likely that some of your students will have heard about Murphy’s Law and some of them will even have seen posters about it. The basic tenet here is that what can go wrong will go wrong. A series of practical situations are explored experimentally to test a number of Murphy’s Laws. The children are encouraged to work out these situations theoretically too. Hence children get a chance to test and explore widely held views under strict, controlled conditions. These experiments are simulations of the real events. Simulating actual events is an important study in itself. Often we cannot run the actual experiment because it would take too long or would be too expensive. So finding a means to simulate the event provides a cheap but accurate way to determine the probability of the event. One of the advantages of computers is that they can simulate quite complicated situations and carry out trials reasonably quickly. One thing that you will have to look out for with simulations done by hand is that the children are scrupulously honest in their experimentation. We have observed children experimenting like this and the probabilities they find turn out to be quite different from the theoretical result. The fact that the children wanted a certain situation to occur influenced, maybe unconsciously, their experimentation. So students have to be encouraged to write down what actually occurred and not what they hoped would occur. This is all part of learning to be as independent an observer as possible, which is an important lesson to learn both for mathematics and for many other areas of life.

sample size, confidence, theoretical probability, experimental estimates of probability, simulation, experimentation

#### Getting Started

- Ask the students if they have heard of Murphy’s Law. Some of them may have heard their parents or other people refer to Murphy’s Law when something annoying has happened to them at a moment of great inconvenience. The following events are examples of what we call Murphy’s Law: being unable to find the car keys when you are in a hurry; getting a flat tyre on your bicycle when you are kilometres from home; rain in the afternoon when you did not take your coat to school (it was sunny when you left home in the morning); and missing the bus because it is early for a change.
- The first lesson will investigate the example of Murphy’s Law relating to keys. It goes like this (act it out as you are saying it):
*There you are carrying a heavy box of things to the door or to the car boot. You put the box in one arm to hold it while you reach inside your pocket for the keys and, you guessed it, the keys are in the other pocket! So you shift the load onto the other arm to get the keys out or you become a contortionist by trying to get it with the opposite arm*. So Murphy’s Law for keys says that keys are always in the pocket that you can’t reach. - Ask the students whether or not they believe in Murphy’s Law for keys and what we might do as an experiment to see if it is true or false. Record their ideas and negotiate a way to trial the event of the box and the keys. As a chance event this can be modelled easily using two cards, one marked key, the other marked Murphy. Without looking the students can put one card in each pocket then carry an empty box to a make believe door. They transfer the load to one arm and reach into their pocket with their free hand. One of two things will occur, they will get the key card or Murphy will strike again!
- Ask the students how many trials they think they will need to check whether Murphy’s Law of keys is correct. The principle of a greater number of trials giving more reliable results should emerge. Each group of students can carry out ten trials and the results from all of these groups can be collated to give a large sample of results. In the trial with the cards the anticipated probability of Murphy’s Law occurring is one half. This can be shown using the table or the tree diagram below. Note that there are four possible outcomes, and the two ticked ones indicate the keys being in the correct pocket.

- There are other factors at work with Murphy’s Law of keys that the students may or may not raise in the class discussion. The majority of people have a preferred hand for picking up things. This hand could also be the one which loads are transferred to so the other hand is free to feel for keys. Such a preference warrants investigation as it could significantly alter the probability of Murphy’s Law of keys occurring. This will have to be done experimentally.

One way is to select people at random and ask them to do these two things:

- Hand them a set of keys and ask them to put them in a pocket;
- Give them a box to carry and ask them to take the box through a door that is shut.

The issue of how many people to ask (sample size) occurs again. Simply put, the more people who are asked, the more confidence we could have in our results. The observations will have to be recorded carefully. A tally chart is a useful tool here.

This observation of people can be done as a homework assignment, if need be. It is wise to tell school students of the need to explain why they are conducting the survey to would-be participants.

Collate the results from as many students or groups as possible. Ask the students to come up with a short report of their findings about Murphy’s Law of keys. The report should include both the theoretical and the experimental results and a conclusion as to the validity of people’s belief in the Law.

#### Exploring

In the middle section of this unit the students will explore some other examples of Murphy’s Law and be encouraged to investigate examples of their own. Below are some examples that can be investigated:

- Murphy’s Law of Traffic Lights
If you are in a rush the traffic lights are always red when you get to them.

This can be simulated by putting two red counters, two green counters, and one orange counter into an ice-cream container and then drawing a counter out at random. This gives you the colour of the lights on your arrival (probability of red is

^{2}/_{5}= 40%). Your degree of haste can easily be simulated by rolling a dice (even numbers for rush, odd for lots of time), making up equal numbers of cards with*rush*and*time*on them and drawing one from a bag each time, or flipping a coin (heads for rush, tails for lots of time).In this case students should observe that your degree of haste makes no difference to the likelihood of getting a red light. This brings up the issue of people believing in Murphy’s Law because unfavourable events in moments of crisis are much more memorable than favourable events in moments of tranquillity.

- Murphy’s Law of Buttered Toast
*The more expensive the carpet the greater the chance that the piece of toast that falls off your plate will land butter side down.*This can be simulated easily using a coin for the piece of toast (heads for butter side, tails for not buttered) and a money dice (faces of 10c, 20c, 50c, $1, 2 & on the blank face students can roll again). Real bread can be used but you need to be culturally sensitive about the use of food in this context. Note that the outcomes of trials could be organised in a chart, like this:

Expense of Carpet Landed 10c 20c 50c $1 $2 Butter 5 2 5 3 7 Not Butter 7 5 3 5 5 Students should realise that in this case the expense of the carpet has no impact on the theoretical probability of the bread landing butter side down. It is people’s recall of unfortunate occurrences that is responsible for this widely accepted corollary of Murphy’s Law.

- Murphy’s Law of Cellotape
*The more of a rush you are in the harder it is to find the start of a roll of cellotape.*Devise your own experiment to test out this version of Murphy’s Law.

- Murphy’s Law of Drawing Pins
*If a drawing pin drops on the floor the chance of it landing sharp end up increases as its distance to the nearest bare foot decreases*.Encourage the children to discuss this and then devise an experiment to test it.

- Murphy’s Law of Supermarket Queues
Whatever queue you join, no matter how short it looks, will always take the longest for you to get served.

This example is more complex in its organisation. It may be easier to have students act it out.

You may wish to discuss with the students what things might affect how long it takes a supermarket queue to get served. They should have recollections of frustrated parents as the person in front took five minutes to write a cheque, or requested an obscure article that took a supermarket employee ages to find, or the checkout operator was in training. Write down the factors that affect the time that a person takes to get through a checkout.

Murphy’s Law of supermarket queues can be simulated in the following way:

Put the students into groups of three. Give each group four ice-cream containers (to be the four checkouts), five blank cubes (with sides labelled 1,2,3,1,2,3, for the number of minutes to serve a customer and the number of new arrivals), twenty cards (ten blank and ten labelled with numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10) to represent the customers.

Model the simulation with one group for the class to see:

Set up the four ice-cream containers with the cubes in them.

*These are the four checkouts at the supermarket*. Put five blank cards (customers) at the first checkout and three at the second.*The manager has noticed that the first two checkouts are busy so she has decided to open the other two. What do you think the waiting shoppers will do?*This will establish the rule that shoppers always go to the shortest queue. Get one student in the group to redistribute the customers (blank cards) among the four checkouts. That student is now responsible for distributing any new arrivals among the queues.A second student is responsible for determining how long it will take to serve each customer. This is done by rolling the checkout dice.

A

*one*means that the customer will take one minute to be served,*two*means two minutes,*three*means three minutes. Get the student to roll the serve times for the first four customers (blank cards). Each dice is left with the serve time face up in the ice-cream container.Tell the students that one-minute has passed. The new arrival’s student rolls the other cube. Whatever number comes up is the number of new customers (cards) arriving. They start by allocating the first few numbered cards to the checkouts with the shortest queues. For example, a three is thrown so the cards 1, 2, and 3 (in order) are placed in checkout queues.

The second student looks at the serve times at each checkout. A customer who had three minutes showing will now have two minutes left so the dice is turned to show

*two*. A customer who had two showing now has one minute left so the dice is turned to show*one*. A customer who had one-minute showing is now through so the third student puts their card in the first minute pile. When a customer goes through the serve time dice is rolled again to decide how long the next customer will take to serve.It is important that the third student collects the customers (cards) as they go through and organises them into successive minute piles. These piles might need to be labelled minute 1, minute 2, etc.

Send the students away to carry out the simulation. Tell them not to touch the minute piles when they are finished, as these will give us the answer to Murphy’s Law of queues. Some students will note the fluctuations that occur in the numbers of customers waiting. Ask the students why this happens (the random effect of the arrivals and serve time dice). Focus on the fact that Murphy’s Law means that people arriving after you will often get served before you. Ask how this could be checked using the minute piles.

This is easily determined by checking which minute pile the successive number cards are in. Start with customer one. Are any later customers in earlier minute piles which mean that they were through the checkout before them? For example, card one might be in the minute four pile and card three might be in the minute two pile so customer three got served before customer one (Murphy strikes again!). Two cards in the same minute pile shows that these customers got through at the same time (Not a Murphy!)

Students can examine each customer number in turn and record the results in a table:

Gather the class together to share the results of the simulation. Focus on the issue,

*Is Murphy’s Law of supermarket queues true or, like the traffic lights, a case of people only remembering when unpleasant things happen?*

#### Reflecting

The final investigation relates to Murphy’s Law of lifts, a curse of the modern office worker. This Law states, *If you are in a hurry to go up or down the lift is never on the floor you are* and *If you get on an lift to go down someone already inside has pressed the button to make it go up*.

- See if the class can come up with an experiment of their own to test this Law. If not you can fall back on the outline below.
- Begin by discussing how lifts work. Important in the description is that the lift responds to cues from the direction buttons that are pushed on different floors of the building. If a button is pressed on floor three the lift travels there to pick up a passenger and stops wherever the person wants to go. It remains on that floor until another button is pressed.

- Tell the students that we are going to simulate Murphy’s Law of lifts using two dice. Our building will have only four floors to make the model simpler. One dice will be thrown to determine which floor a passenger is beginning their journey on. In the morning people enter an lift on the first floor since this is the level of entry into the building and they are going to their office. The beginning floor dice is labelled 1,1,1,2,3,4. The destination dice is labelled differently for two reasons:

- people staying on floor one don’t use the lift;
- the lower the floor they are going to the more likely they are to use the stairs and not the lift.

The destination dice is labelled 2,3,3,4,4,4.

- To carry out the simulation each group of students will need a beginning dice and a destination dice. They will also need to draw a building with four floors and a counter to be the lift.
The simulation proceeds like this:

Roll the beginning dice to determine the starting floor for passenger zero (do not record their result). Roll the destination dice to find out which floor they take the lift to. Leave the lift (counter) on that floor.

Now roll the same dice to get the starting and destination floor for passenger one.

Do they have to wait for the lift? Record the results in a tally chart.

Continue like this, until thirty passengers have travelled on the lift.

- Before they begin the simulation ask the students to predict how often a passenger will need to wait for the lift and how often it will be on the floor they start from. Get them to justify their reasoning preferably using theoretical models such as tables that have been used elsewhere in the unit.
- Once the simulation is complete get them to compare the results with their theoretical predictions and to comment on any discrepancies. A possible model for the simulation is shown in the table below. Note that only one sixth of the time a passenger does not have to wait. W = wait N = no wait
Destination Dice

2

3

3

4

4

4

Starting

Dice

1

W

W

W

W

W

W

1

W

W

W

W

W

W

1

W

W

W

W

W

W

2

N

W

W

W

W

W

3

W

N

N

W

W

W

4

W

W

W

N

N

N

Discuss with the students how the makers of the lift could improve this situation. One way is to programme the lift to return to the ground floor when empty. This will cut down on waiting time but will it result in more wear on the lift?

- Ask the students to suggest what dice should be used to model the likely starting floors and destination floors for the afternoon given that during that time many workers will leave their offices for the first floor to go home. They can simulate this scenario in the same way and suggest how the lift designers could modify the situation to decrease the occurrence of waiting. One possibility is to get the lift to return to floor two or three when unused. What impact would this have on wait occurrence and time, and wear on the lift?

Dear Parents and Whanau,

At school this week we have been exploring Murphy’s Law. The basic idea behind Murphy’s Law is that what can go wrong will go wrong. We have investigated Murphy’s Law of:

- Buttered toast
- Supermarket queues
- Lifts
- Keys in pockets
- Traffic Lights

### Home Activity

### Murphy’s Law -

A simple example of Murphy’s Law operating in sport is some Captains’ total inability to win the coin toss.

At home this week you are to try as hard as you can to emulate the feat of a certain New Zealand cricket captain by losing the toss four times in a row. Investigate with your child;

- How easy is this to do?
- Experiment to find out.
- Write down the results of your experiment and write a statement about how easy it is to lose the toss four times in a row.

Discuss together how Murphy's Law operated for you.

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