Transcript 2.6 Normal Probability and Simulations

Simulations and Probability
An Internal Achievement
Standard worth 2 Credits
Probability
Simulations
Normal Distribution
Jim on the Edge
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
Trial
Off
Edge
1
2
3
4
5
1
Result
Safe
2
Fallen
3
4
5
6
7
8
Random Number from calculator
2231233311
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
Results for each Student
Student 1
Student 2
Student 3 Student 4
Student 5
1
2
3
4
5
6
7
8
9
10
11
12
Summary Table
Successful Shots
Tally Chart
Number of Students
Probability
0 1 2 3 4 5 6 7 8 9 10 11 12
Student 6 …
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
Normal Curve
0.6
0.5
Proportions
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
1
2
3
-0.1
Standard Deviations from the mean
Key Points
– The mean is in the middle
– It’s Symmetrical
– The Standard Deviation determines the width
4
Normal Distribution
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.
1 SD
2 SD’s
3 SD’s
μ
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 13.
13
P = 50% 10
1 S.D. so ½ of 68%
= 34%
P (number less than 13)
= 50% + 34%
= 84%
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 ‘zvalues’.
– 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 = 70.4kg
μ = 70.4
Std Dev = 4.8kg
σ = 4.8
What is the probability that a student weighs over 68
kg. x = 68
z = (68 – 70.4)
4.8
= -0.5
Then we would look up -0.5 in the standard normal
tables.
Reading Tables
z
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
So P = 0.1915
Read down to 0.5
and across to 0
…
…
Answering Questions
Start with a diagram
68 70.4
Shaded area will tell us the probability we want
to find…
P = 0.1915
Table told us
P = 0.5
We know
So P(weight > 68)
= 0.5 + 0.1915
= 0.6915
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
z
x

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
Red Area = 0.5 – 0.35
70.4
= 0.150
Blue Area = 0.35
Find Z
Look up 0.1500 in the middle of the table, read back to z.
z = ±0.385, (minus because it’s the left hand side)
Find unknown using
x
z

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.
Normal C. D
Lower
:0
Upper
:0
σ
:0
μ
:0
Execute
F1
F2
F5
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.
Inverse Normal
Area
:0
σ
:0
μ
:0
Execute
F1
F3
F5