Simulations and Probability An Internal Achievement Standard worth 2 Credits

Probability Simulations Jim on the Edge Normal Distribution Normal Curve Standardised Values Reading Tables Practice Example Inverse Normal

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

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

Simulations Jim on the Edge Jim starts standing on the edge of a cliff He has a 1/3 chance of stepping off or towards the edge of the cliff and a 2/3 chance of stepping away from the edge. If he gets 5 steps away from the edge he is safe. Write and do a simulation to work out his chances of not falling off the edge of the cliff.

Simulations Description Generate random numbers on a calculator from 1 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. Continue to generate numbers till he has either fallen, or is safe (5 steps from the edge). Repeat 10 times and record how many times he falls and how many times he is safe. The probability of being safe is: Number of times safe Total number of trials

Simulations Simulation Off1Edge4325TrialResult Random Number from calculator 213 Safe Fallen

Simulations Now lets use the simulation to work some things out … 1)Jim walks home this way every night of the week. How often does he go over the edge? 2)Over the course of a month (31 days), how often will Jim not fall over the edge?

Simulations Remember the key points to writing a simulation are; –What are the Proportions? –How do you generate them? –What do you record? –What do you do with the values you record? –How do you find the probability required?

Basketball Simulation A group of 20 Students are introduced to basketball. History shows that, after a short practice, students are successful with 30% of their free throws. Each student is given 12 free throws. You are going to carry out a simulation to estimate the number of successful throws for each student. Carry out the simulation for each student a)Describe a simulation using random numbers to model the simulation above b)Record your results for each student in a table (shown next page) c)Summarise your results in a table (shown next page)

Basketball Simulation Student 1Student 2Student 3Student 4Student 5Student … Results for each Student Summary Table Successful Shots Tally Chart Number of Students Probability

Basketball Simulation Some questions Based on your simulation, what is the number of shots that a student is most likely to get out of 12? 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 at least 50% of their shots in?

Simulations Practice –Homework Book P105 to 110

Normal Distribution Normal distribution is a reasonably common probability distribution. It assumes that most data is around the average and extreme cases are less likely but symmetrically distributed on either side of the mean. When graphed it looks like this…

The Normal Curve Key Points –The mean is in the middle –It’s Symmetrical –The Standard Deviation determines the width

It is the area under the Standard Normal Curve that we use to calculate the probabilities. –Eg) Area under half of the curve = ½ of 1 = 0.5 So the probability that our value is greater than the mean is 0.5 or 50%. Normal Distribution

In fact for all normal distributions. μ 1 SD 3 SD’s 2 SD’s P ≈ 68% P ≈ 95% P > 99%

Normal Distribution Eg) Mean (μ) = 10 Standard Deviation (S.D.) = 3 Find the probability that the number picked is less than P = 50% so ½ of 68% = 34% P (number less than 13) = 50% + 34% = 84% 1 S.D.

Normal Curve Practice P350 Exercise 30.1 (Work through 1 &2 together) P353 Exercise 30.2 Q 1 & 2 together then odds Homework P111 and 112

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

Using Normal Distribution So a method of standardising data was created called standard approximation –If our data value is x our mean in is μ our Standard Deviation is σ We can standardise the data value using the formula below z = x – μ σ Then look up z in the standard normal tables

Standard Approximation Eg) Y12 Student weights mean = 70.4kgμ = 70.4 Std Dev = 4.8kgσ = 4.8 What is the probability that a student weighs over 68 kg. x = 68 z = (68 – 70.4) = -0.5 Then we would look up -0.5 in the standard normal tables.

Reading Tables z …12345… … … … … … … … 4811… … 4711… … 3710… … 3610… … 369… … 36… … 35… … 2… … 2… … 2… … 2… … 1… … Finding z = Read down to 0.5 and across to 0 So P =

Answering Questions Start with a diagram Shaded area will tell us the probability we want to find… Table told us We know P = P = 0.5 So P(weight > 68) = = Table always tells us the mean to the z-value.

Practice If the mean is 70.4kg and the Standard Deviation is 4.8kg a) Probability the weight is less than 76kg b) Probability the weight is more than 80kg c) Probability the weight is between 56kg and 71kg Step by step Draw normal curve picture, shade what you are finding Write the values you know, mean, SD, X-value Calculate the Z-value Look up Table Answer Question

Normal Distribution Practice P359 Exercise 30.4 All, Do Q1 and 2 together Homework P112 to 113

Inverse Normal These are problems where you are given the Probability and asked to find x, μ or σ. Eg) Mean weight of Y12 boys is 70.4kg, SD is 4.8kg. A new rugby grade is being established for the lightest 35% of Y12 boys, calculate the weight limit. Draw a picture Find Z Look up in the middle of the table, read back to z. z = ±0.385, (minus because it’s the left hand side) Find unknown using 70.4 Blue Area = 0.35 Red Area = 0.5 – 0.35 = 0.150

Inverse Normal Practice Page 366 exercise 30.6 Homework P115 to 116

On a Graphics Calculator These problems can all be simplified using a Graphics Calculator Most Problems –In Stats Mode Dist – F5 Norm – F1 Ncd – F2 The upper and lower limits depend on the context of the problem, and the shaded area. If no lower is given use a very low number (often 0). If no upper is given use a very large number, compared to mean and SD. F5 Normal C. D Lower: 0 Upper: 0 σ: 0 μ: 0 Execute F2F1

On a Graphics Calculator Inverse Normal Problems –In Stats Mode Dist – F5 Norm – F1 InvN – F3 The calculator differs from the tables here as it always uses the area left of the x-value. F5 Inverse Normal Area: 0 σ: 0 μ: 0 Execute F3F1