Simulations and ProbabilitySimulations and Probability

An Internal Achievement An Internal Achievement Standard worth 2 CreditsStandard worth 2 Credits

ProbabilityProbability

Simulations

Jim on the Edge

Normal Distribution

Normal Curve

Standardised Values

Reading Tables

Practice Example

Inverse Normal

SimulationsSimulations

These are the same as last year, A These are the same as last year, A situation is described and you have to situation is described and you have to write a description of how you would do write a description of how you would do the simulation stating clearly;the simulation stating clearly;– The proportions and how to generate themThe proportions and how to generate them– What you recordWhat you record– What you do with the values you recordWhat you do with the values you record– How you find the probability requiredHow you find the probability required

SimulationsSimulations

The new part about simulations is that you The new part about simulations is that you actually do the simulation this year, record actually do the simulation this year, record the results and make some calculations the results and make some calculations based on those results.based on those results.

SimulationsSimulations

Jim on the EdgeJim on the EdgeJim starts standing on the edge of a cliffJim starts standing on the edge of a cliff

He has a 1/3 chance of stepping off or towards the He has a 1/3 chance of stepping off or towards the edge of the cliff and a 2/3 chance of stepping away edge of the cliff and a 2/3 chance of stepping away from the edge.from the edge.

If he gets 5 steps away from the edge he is safe.If he gets 5 steps away from the edge he is safe.

Write and do a simulation to work out his chances Write and do a simulation to work out his chances of not falling off the edge of the cliff.of not falling off the edge of the cliff.

SimulationsSimulations

DescriptionDescriptionGenerate random numbers on a calculator from 1 Generate random numbers on a calculator from 1 to 3, if it’s a 1 he moves towards the edge, if it’s a to 3, if it’s a 1 he moves towards the edge, if it’s a 2 or a 3 he moves away from the edge.2 or a 3 he moves away from the edge.

Continue to generate numbers till he has either Continue to generate numbers till he has either fallen, or is safe (5 steps from the edge).fallen, or is safe (5 steps from the edge).

Repeat 10 times and record how many times he Repeat 10 times and record how many times he falls and how many times he is safe.falls and how many times he is safe.

The probability of being safe is:The probability of being safe is:Number of times safeNumber of times safe

Total number of trialsTotal number of trials

SimulationsSimulations

SimulationSimulationOff 1Edge 432 5Trial Result

1

2

3

4

5

6

7

8

Random Number from calculator 2 13

Safe

Fallen

2 3 32 3 1 1

SimulationsSimulations

Now lets use the simulation to work Now lets use the simulation to work some things out …some things out …

1)1) Jim walks home this way every night of the Jim walks home this way every night of the week. How often does he go over the edge?week. How often does he go over the edge?

2)2) Over the course of a month (31 days), how Over the course of a month (31 days), how often will Jim often will Jim notnot fall over the edge? fall over the edge?

SimulationsSimulations

Remember the key points to writing a Remember the key points to writing a simulation are;simulation are;

– What are the Proportions?What are the Proportions?– How do you generate them?How do you generate them?– What do you record?What do you record?– What do you do with the values you record?What do you do with the values you record?– How do you find the probability required?How do you find the probability required?

Basketball SimulationBasketball Simulation

A group of 20 Students are introduced to basketball. A group of 20 Students are introduced to basketball. History shows that, after a short practice, students are History shows that, after a short practice, students are successful with 30% of their free throws.successful with 30% of their free throws.

Each student is given 12 free throws.Each student is given 12 free throws.

You are going to carry out a simulation to estimate the You are going to carry out a simulation to estimate the number of successful throws for each student. number of successful throws for each student.

Carry out the simulation for each studentCarry out the simulation for each student

a)a) Describe a simulation using random numbers to model the Describe a simulation using random numbers to model the simulation abovesimulation above

b)b) Record your results for each student in a table (shown next Record your results for each student in a table (shown next page)page)

c)c) Summarise your results in a table (shown next page)Summarise your results in a table (shown next page)

Basketball SimulationBasketball Simulation

Student 1 Student 2 Student 3 Student 4 Student 5 Student 6123456789101112

…

Results for each Student

Summary Table

Successful Shots

Tally Chart

Number of Students

0 21 43 65 7 111098 12

Probability

Basketball SimulationBasketball Simulation

Some questionsSome questionsBased on your simulation, what is the number of Based on your simulation, what is the number of shots that a student is most likely to get out of 12?shots that a student is most likely to get out of 12?

If you were to repeat this with 100 students, If you were to repeat this with 100 students, – How many would you expect to get 3 out of 12 shots in?How many would you expect to get 3 out of 12 shots in?– How many would you expect to get at least 50% of their How many would you expect to get at least 50% of their

shots in?shots in?

SimulationsSimulations

PracticePractice– Homework Book P105 to 110Homework Book P105 to 110

Normal DistributionNormal Distribution

Normal distribution is a reasonably Normal distribution is a reasonably common probability distribution. common probability distribution.

It assumes that most data is around the It assumes that most data is around the average and extreme cases are less likely average and extreme cases are less likely but symmetrically distributed on either side but symmetrically distributed on either side of the mean.of the mean.

When graphed it looks like this…When graphed it looks like this…

The Normal CurveThe Normal Curve

Key PointsKey Points– The mean is in the middleThe mean is in the middle– It’s SymmetricalIt’s Symmetrical– The Standard Deviation determines the widthThe Standard Deviation determines the width

Normal Curve

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-4 -3 -2 -1 0 1 2 3 4

Standard Deviations from the mean

Pro

po

rti

on

s

It is the area under the Standard It is the area under the Standard Normal Curve that we use to calculate Normal Curve that we use to calculate the probabilities.the probabilities.

– Eg) Area under half of the curve Eg) Area under half of the curve

= ½ of 1= ½ of 1

= 0.5= 0.5

So the probability that our value is greater than the So the probability that our value is greater than the mean is 0.5 or 50%.mean is 0.5 or 50%.

Normal DistributionNormal Distribution

Normal DistributionNormal Distribution

In fact for all normal distributions.In fact for all normal distributions.

μ

1 SD

3 SD’s2 SD’s

P ≈ 68%

P ≈ 95%P > 99%

Normal DistributionNormal Distribution

Eg) Eg) Mean (Mean (μμ) = 10) = 10

Standard Deviation (S.D.) = 3Standard Deviation (S.D.) = 3

Find the probability that the number Find the probability that the number picked is less than 13.picked is less than 13.

10 13P = 50%

so ½ of 68%

= 34%

P (number less than 13) = 50% + 34%

= 84%

1 S.D.

Normal CurveNormal Curve

PracticePracticeP350 Exercise 30.1 (Work through 1 &2 together)P350 Exercise 30.1 (Work through 1 &2 together)

P353 Exercise 30.2 Q 1 & 2 together then oddsP353 Exercise 30.2 Q 1 & 2 together then odds

Homework P111 and 112Homework P111 and 112

Standardised ValuesStandardised Values

In theory this seems ideal and because all In theory this seems ideal and because all normal curves are of a similar shape (the normal curves are of a similar shape (the bell curve) the data can be standardised, bell curve) the data can be standardised, these standardised values are called ‘z-these standardised values are called ‘z-values’.values’.– Mean of zMean of z = 0= 0– SD of zSD of z = 1= 1– Area under a standardised curve = 1 or 100%Area under a standardised curve = 1 or 100%

Using Normal DistributionUsing Normal Distribution

So a method of standardising data was created So a method of standardising data was created called standard approximationcalled standard approximation– If If our data value is our data value is xx

our mean in is our mean in is μμ

our Standard Deviation is our Standard Deviation is σσ

We can standardise the data value using the formula belowWe can standardise the data value using the formula below

zz = = xx – – μμ σσ

Then look up Then look up zz in the standard normal tables in the standard normal tables

Standard ApproximationStandard Approximation

Eg) Y12 Student weightsEg) Y12 Student weights

mean = 70.4kgmean = 70.4kg μμ = 70.4 = 70.4

Std Dev = 4.8kgStd Dev = 4.8kg σσ = 4.8 = 4.8

What is the probability that a student weighs over 68 What is the probability that a student weighs over 68 kg. kg. x = 68x = 68

z z = = (68 – 70.4)(68 – 70.4)

4.84.8

= -0.5= -0.5

Then we would look up -0.5 in the standard normal Then we would look up -0.5 in the standard normal tables.tables.

Reading TablesReading Tablesz 0 1 2 3 4 5 6 … 1 2 3 4 5 …

0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 … 4 8 12 16 20 …

0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 …   4 8 12 16 20 …

0.2 0.0793 0.0832 0.0871 0.0910 0.0948 …   4 8 12 15 …  

0.3 0.1179 0.1217 0.1255 0.1293 …   4 8 11 …  

0.4 0.1554 0.1591 0.1628 0.1664 …   4 7 11 …  

           

0.5 0.1915 0.1950 0.1985 …   3 7 10 …  

0.6 0.2258 0.2291 0.2324 …   3 6 10 …  

0.7 0.2580 0.2612 0.2642 …   3 6 9 …  

0.8 0.2881 0.2910 …   3 6 …    

0.9 0.3159 0.3186 …   3 5 …    

           

1.0 0.3413 …   2 …      

1.1 0.3643 …   2 …      

1.2 0.3849 …   2 …      

1.3 0.4032 …   2 …      

1.4 0.4192 …   1 …      

…                            

Finding z = -0.500 Read down to 0.5and across to 0

So P = 0.1915

Answering QuestionsAnswering Questions

Start with a diagramStart with a diagram

Shaded area will tell us the probability we want Shaded area will tell us the probability we want to find…to find…

Table told us Table told us

We knowWe know

68 70.4

P = 0.1915P = 0.1915

P = 0.5P = 0.5

So P(weight > 68) So P(weight > 68) = 0.5 + 0.1915= 0.5 + 0.1915= 0.6915= 0.6915

Table always tells us the mean to the z-value.Table always tells us the mean to the z-value.

PracticePractice

If the mean is 70.4kg and the Standard If the mean is 70.4kg and the Standard Deviation is 4.8kgDeviation is 4.8kg

a)a) Probability the weight is less than 76kgProbability the weight is less than 76kg

b)b) Probability the weight is more than 80kgProbability the weight is more than 80kg

c)c) Probability the weight is between 56kg and 71kgProbability the weight is between 56kg and 71kg

Step by stepStep by stepDraw normal curve picture, shade what you are findingDraw normal curve picture, shade what you are finding

Write the values you know, mean, SD, X-valueWrite the values you know, mean, SD, X-value

Calculate the Z-valueCalculate the Z-value

Look up TableLook up Table

Answer Question Answer Question

xz

Normal DistributionNormal Distribution

PracticePracticeP359 Exercise 30.4 All, Do Q1 and 2 togetherP359 Exercise 30.4 All, Do Q1 and 2 together

Homework P112 to 113Homework P112 to 113

Inverse NormalInverse Normal

These are problems where you are given the These are problems where you are given the Probability and asked to find Probability and asked to find xx, , μμ or or σσ..

Eg) Mean weight of Y12 boys is 70.4kg, SD is 4.8kg.Eg) Mean weight of Y12 boys is 70.4kg, SD is 4.8kg.

A new rugby grade is being established for the lightest A new rugby grade is being established for the lightest 35% of Y12 boys, calculate the weight limit.35% of Y12 boys, calculate the weight limit.Draw a pictureDraw a picture

Find ZFind Z

Look up 0.1500 in the middle of the table, read back to z.Look up 0.1500 in the middle of the table, read back to z.

z = z = ±0.385, ±0.385, (minus because it’s the left hand side)(minus because it’s the left hand side)

Find unknown usingFind unknown using xz

70.4Blue Area = 0.35

Red Area = 0.5 – 0.35 = 0.150

Inverse NormalInverse Normal

PracticePracticePage 366 exercise 30.6Page 366 exercise 30.6

Homework P115 to 116Homework P115 to 116

On a Graphics CalculatorOn a Graphics CalculatorThese problems can all be These problems can all be simplified using a Graphics simplified using a Graphics CalculatorCalculator

Most ProblemsMost Problems– In Stats ModeIn Stats Mode

Dist – F5Dist – F5

Norm – F1Norm – F1

Ncd – F2Ncd – F2The upper and lower limits depend on the The upper and lower limits depend on the context of the problem, and the shaded area. context of the problem, and the shaded area. If no lower is given use a very low number If no lower is given use a very low number (often 0).(often 0).

If no upper is given use a very large number, If no upper is given use a very large number, compared to mean and SD.compared to mean and SD.

F5

Normal C. DLower : 0Upper : 0σ : 0μ : 0Execute

F2F1

On a Graphics CalculatorOn a Graphics Calculator

Inverse Normal ProblemsInverse Normal Problems– In Stats ModeIn Stats Mode

Dist – F5Dist – F5

Norm – F1Norm – F1

InvN – F3InvN – F3

The calculator differs from the tables The calculator differs from the tables here as it always uses the area left of here as it always uses the area left of the x-value.the x-value.

F5

Inverse NormalArea : 0σ : 0μ : 0Execute

F3F1