SIMULATIONS2.13 Probability

The Idea:

A simulation uses Maths to imitate a real situation

It’s supposed to give similar results

..and so acts as a predictor of what should actually happen in real life

It is a model in which repeated experiments are carried out for the purpose of estimating in real life

We use a mathematical tool to generate numbers, which we map onto real life to estimate results cheaply and quickly.

Simulations…

Are used to solve problems using experiments when it is difficult to calculate theoreticallyOften involve either the calculation of:• The long-run relative frequency of a

successful event (probability of success)• The average number of ‘visits’ taken to

gain a ‘full-set’ (I need to make 6 visits)Sometimes have limits put on them – how many can I get for $14? How often will I be successful?

What are your chances..?

1. If you had a baby, how likely is it that you would get a girl?

2. If you threw a dice, that you would get a 6?

3. That I am going to beat up Marcel before the hour is out?

4. That the next 4 days will be sunny?What assumptions did you just

make?!

Things to remember

We are using maths tools to pretend or “simulate” the real situation. That way we can predict the outcome without actually having to be in the real life situation.

Things to remember

1. Tool2. Trial3. Assumptions4. Results5. Calculations6. Conclusion

TOOL• This is how we decide what we will simulate

the actual situation with.

• There are many different types of tools available to us: • Coins

• Dice• Spinners• Random number generators• And many others…

• They all have one thing in common: they are fair.

• When we describe how we use the tool, we must be specific – write for an idiot!!!!

TOOL1. We are using Ran# on the calc.2. For 5 items, numbered 10, 11…14

we would put in DATA Ran# + STARTER ie 5 Ran# + 10.3. Always write ‘ignore decimals’ if

appropriate.We must assume the person knows nothing

about Mathematics when we write the

Tool-i.e. be veryspecific.

TOOLWhich tool would you use to:-1. Determine the probability of having

3 boys (sex of the child)2. Find out how many rainy days we

should have had this year? P(rain)=0.2

3. Work out whether an engine on your plane was going to fail . 4 engines, P(fail)=40%

TRIAL•Define what ONE trial is.•Decide how we will stop each trial•What designates a SUCCESS in the trial.•We also have to decide HOW MANY trials are required to be statistically significant-we usually do between 20 and 30.

The Six Dwarfs

Description

A cereal company wants to promote its product. Inside each box of cereal is one dwarf – Sneezy, Happy, Grumpy, Bashful, Doc, and Sleepy. There are equal numbers of each dwarf.

When you have collected all six dwarfs, you can send away for a life-sized version of the seventh dwarf, Dopey.

Task

Calculate how many boxes of cereal you need to buy on average to collect all the dwarfs and get Dopey.

ASSUMPTIONS

• All events are independent (the outcome of one event won’t affect the other)•We have fair tools (ie the coin won’t be weighted to come up heads)• If we did the experiment again, we would probably come up with a slightly different estimate

RESULTS•We always put our results into a table as then they are easy to use in the calculations. (Must show you are organised!)•Often the table is drawn for you, but sometimes you will have to draw your own.• Put in a ‘success’ column if needed.

CALCULATION• Always answer the question!

• This is often an average (mean) or a proportion/probability:

Mean = number of cereal boxes needed total number of trials

Prob =favourable

total

• We can also answer expected number (‘how many’)questions - probability multiplied by number of trials.

This will NEVER be bigger than 1 or 100%!!! Just sayin’..

CONCLUSIONYou must state the following at some

stage in your simulation:My result is only an estimate. If I

repeated this simulation, I would probably get different results and my estimate would be different.

ANSWER the question or problem! Continued ….

CONCLUSION ctd

Give both the maths answer (5.16 – show working!) and the real life answer (I would need to buy at least 5 boxes of cereal to get all 4 animals. To be on the safe side, I would advise someone to buy six boxes, as the mathematical answer is over 5.)

Assumptions – you assume that the tool is FAIR, that the selection is RANDOM and that each thing has an equal chance of being chosen.

ELI – effects, limitations, improvements – link to the real life situation and say how each would affect your simulation.

WHATIFS!!!!

ELI – effects, limitations, improvements – link to the real life situation and say how each would affect your simulation.

Whatif…. Come up with a few ideas – use your imagination! THEN put your Maths Hat on and say how it would change or affect your simulation or results or sampling method or… something

Whatif… someone STOLE all the Goofies? Whatif .. The machine broke and put in no

Spongebobs?Whatif.. The black dye ran so more Mickeys were

discarded?

Merit – TTRCA +explain +assumptions

• Tools – link to context and explain why you allocated certain numbers• Trial• Results – explain how you will record results in the table• Calculation – use your results to justify your findings• Answer – Explain what would happen if you were to do another simulation

- write about assumptions

Excellence – Merit + effects + improvement

• Tools – link to context and explain why you allocated certain numbers• Trial – clearly explain success, linking to context• Results – explain how you will record results in the

table• Calculation – use your results to justify your findings• Answer – Explain what would happen if you were to

do another simulation- write about assumptions and their effect on the simulation- clearly explain how to improve the simulation to better reflect the context

Practice Simulation

• We are in Disneyland and the promoters are running a Donald Duck promotion.• Every time you pay for a ride you collect a

card with a picture of one of Donald Duck’s nephews-Huey, Duey or Louie. • P(Huey)=0.5, P(Duey)=0.2, P(Louie)=0.3• When you collect all 3 Ducks, you win a BIG

Donald Duck.

Simulate the situation and determine how many rides you need to go on to collect all 3 ducks and therefore collect your BIG DUCK prize.Remember to follow the rules of

1.Tool2.Trial3.Assumptions4.Results5.Calculation

Your second practice

•What is the probability that a 3 child family will contain exactly 3 boys? You may assume that a boy or girl is equally likely to be born.

Links and investigations

Tools• Dice Roller

• Throwing coins

• Drawing playing cards

• Spinners

• Random number generator

Sites• Monty Hall – my favourite!

• Try the cereal box simulation here

• Birthdays - The answer may surprise you!

Think about this:

Miss Fuller creates the following cards:

You choose a card each time you enter a maths class on time.

When you have collected cards to spell HAVE FUN you may take the afternoon off.

How many visits, on average, will it take you?

OOPS – now do it again. BAD Miss Fuller!!

H A V E F U N N N N