probability – risk-free
DESCRIPTION
Lars Thomsen, Avondale College . Probability – Risk-Free. What has changed?. Re- conceptualising probability. Changes in the Probability Standard. Old 2.6. Simulate probability situations. Apply the normal distribution. Tree diagrams (old Level 1 probability). New 2.13 Simulation . - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/1.jpg)
PROBABILITY – RISK-FREELars Thomsen, Avondale College
![Page 2: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/2.jpg)
Re-conceptualising probability
What has changed?
![Page 3: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/3.jpg)
Changes in the Probability Standard
Old 2.6
New 2.13Simulation
Tree diagrams(old Level 1 probability)
Simulate probability situations
Apply the normal distribution
Risk and relative risk (new)
New 2.12Probability
Experimental distributions(progression from new 1.13)
![Page 4: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/4.jpg)
2.12 Probability
evaluate statistically based reports interpreting risk and relative risk
investigate situations that involve elements of chance
comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions
calculating probabilities, using tools such as two-way tables, tree diagrams
![Page 5: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/5.jpg)
2.12 Probability
Methods include a selection from those related to:
risk and relative risk the normal distribution experimental distributions relative frequencies two-way tables probability trees.
![Page 6: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/6.jpg)
From the T&L guides:
IndicatorsA. Comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions: Describes and compares distributions and
recognises when distributions have similar and different characteristics.
Carries out experimental investigations of probability situations …
Is beginning to use mean and standard deviation as sample statistics or as population parameters.
Chooses an appropriate model to solve a problem.
![Page 7: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/7.jpg)
From the T&L guides:
IndicatorsB. Calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology: Uses two-way frequency tables to solve simple
probability problems ... Constructs and interprets probability trees ... Students learn that situations involving real
data from statistical investigations can be investigated from a probabilistic perspective .
![Page 8: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/8.jpg)
From the T&L guides:
IndicatorsB. Calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology: Uses two-way frequency tables to solve simple
probability problems ... Constructs and interprets probability trees ... Students learn that situations involving real
data from statistical investigations can be investigated from a probabilistic perspective .
![Page 9: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/9.jpg)
Re-conceptualising probability
Theoretical probability vs. Experimental probability
Probability as a way of making sense of data
from
to
• Similarities to statistical methods• But: variation through chance, not
sampling
![Page 10: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/10.jpg)
Re-conceptualising probability
Statistics Probability Sample Population Variability from
sampling Median / IQR (or
others) Inference Continuous variable
Experimental distribution
Infinite / not defined? Variability from
chance Mean / sd Model Continuous
(histogram) or discrete / categorical ( two way table) variable
![Page 11: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/11.jpg)
Experimental distribution or not? Test reaction time of all students in
class Take a sample of 50 reaction times
from C@S Compare two samples of 50 reaction
times from C@S Plant 30 sunflowers and measure the
height after 2 weeks Measure the length of the right foot
of each student in class (in winter) DISCUSS!
![Page 12: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/12.jpg)
What we have learnt from AS91038
Introducing probability distributions
![Page 13: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/13.jpg)
LO: Carry out an investigation involving chance
INVESTIGATIONS INTO CHANCE
DICE BINGO
![Page 14: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/14.jpg)
DICE BINGO!Fill out a 3x3 grid with numbers from 0-5. You may use numbers more than once. E.g.
To play dice bingo two dice are rolled and the difference between them is the number called out. E.g. = 3
The winner is the first to get the whole grid.
LO: Carry out an investigation involving chance
3 0 25 4 14 3 0
![Page 15: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/15.jpg)
PROBLEMWrite down an appropriate problem statement for this investigation
Write down what you think the answer will be
Write down how you think we could investigate this problem
LO: Carry out an investigation involving chance
![Page 16: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/16.jpg)
PLANI roll 2 dice and work out the difference between the numbers, I will do this 30 times
I will draw up a table from 1 – 30 and write down the difference of the two dice each time
I will also draw up a tally chart to keep a track of how many times I get 0, 1, 2, 3, 4, or 5 as the difference of the two dice
LO: Carry out an investigation involving chance
![Page 17: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/17.jpg)
DATA
LO: Carry out an investigation involving chance
Trial Difference of the two dice
123
30Difference of the two dice
Tally Frequency
012345
![Page 18: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/18.jpg)
ANALYSISDraw a graph of the difference between the two dice against the trial number
LO: Carry out an investigation involving chance
5 -4 -3 -2 -1 -0
Difference of the two dicevs trial number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 30
Trial Difference of the two dice
1 22 33 04 5
e.g.
Trial number
Diffe
renc
e of
the
two
dice
![Page 19: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/19.jpg)
ANALYSISDescribe what you see in your graph:Draw a horizontal line where you think the outcomes jump around and link this the typical outcome
Identify any “runs” of outcomes and link this with whether each trial outcome appears to be independent
Identify the range of the outcomes and link this with the possibilities for the outcomes
LO: Carry out an investigation involving chance
![Page 20: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/20.jpg)
ANALYSISDraw a dot plot for frequency of the difference of the two dice
LO: Carry out an investigation involving chance
Dot plot of difference of the two dice
e.g.
0 1 2 3 4 5 Difference of the two dice
Freq
uenc
y
Difference of the two dice
Tally Frequency
0 | 11 ||| 3
![Page 21: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/21.jpg)
ANALYSISDescribe what you see in your dot plot: Identify the “tallest” outcome and link this with the mode “most common”
Circle the “towers” that represent at least 50% of the outcomes and link these with “most likely”
Outline the shape of the dot plot and link this with the shape of the distribution (skewed, symmetric, bi-modal)
LO: Carry out an investigation involving chance
![Page 22: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/22.jpg)
ANALYSISCreate a distoplot by drawing rectangles around your dots LO: Carry out
an investigation involving chance
Distoplot of number of heads
0 1 2 3 4 5 6 Number of heads
Freq
uenc
y
![Page 23: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/23.jpg)
ANALYSISWork out the probability of getting each of the outcomes and write at the top of the box
LO: Carry out an investigation involving chanceDistoplot of number of
heads
0 1 2 3 4 5 6 Number of heads
Freq
uenc
y 3/30 = 0.10.1 = 10%
10%
20%
33%
10%3%
23%
![Page 24: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/24.jpg)
CONCLUSIONWrite an answer to your problem and provide supporting evidence from your investigation:Clearly give an answer “Based on my experiment, I would estimate that….”
What are you pretty sure about what do you think (would be the same with another experiment and why)?
What are you not so sure about (what do you think would change with another experiment and why)?
LO: Carry out an investigation involving chance
![Page 25: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/25.jpg)
Snail Race
http://www.transum.org/software/SW/SnailRace/
![Page 26: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/26.jpg)
PROBLEMIf I flip 6 coins, how many heads will I get?
Write down what you think the answer will be
Write down how you think we could investigate this problem
LO: Graph and describe a probabi l i ty distribut ion
![Page 27: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/27.jpg)
PROBLEM:Jessica thinks she’s really good at archery and tells her friends that she can get a bull's-eye 4 out of 5 shots. Her friends want to find out the truth.
NOW YOU…Write down an appropriate problem statement for this investigation
Write down how you think we could investigate this problem
LO: Write problem statements, analysis statements and conclusions
![Page 28: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/28.jpg)
Progression from Y11
1.13 Probability Investigation 2.12 Probability Discrete data
Experiment ‘Distoplot’ ‘Rough shape’
Proportion
(Discrete and) continuous data
Experiment Histogram ‘Rough shape’, skew,
‘peakedness’ Proportion, mean and
sd
![Page 29: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/29.jpg)
Making sense of standard deviation
![Page 30: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/30.jpg)
Investigation: age estimation Estimate the age of
this gentleman at the time the picture was taken.
How good are we? Justify your
answer!!!
![Page 31: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/31.jpg)
Accuracy or Consistency?
Which measures do describe the two?
![Page 32: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/32.jpg)
Accuracy or Consistency?
Add 10 shots for a shooter who is consistently bad.
Draw a histogram Calculate mean and sd
![Page 33: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/33.jpg)
Accuracy or Consistency?
Add 10 shots for a shooter who is inconsistently average.
Draw a histogram Calculate mean and sd
![Page 34: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/34.jpg)
Accuracy or Consistency?
Add 10 shots for a shooter who is consistently great.
Draw a histogram Calculate mean and sd
![Page 35: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/35.jpg)
Comparing experimental distributions with the normal distribution
![Page 36: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/36.jpg)
Features of the normal distribution
Continuous random variable
Bell shape Symmetric
about μ
![Page 37: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/37.jpg)
How normal is normal?
1
2
3
4
5
6
7
8
9
16 18 20 22 24 26 28 30 32
2
4
6
8
10
12
0 10 20 30 40
2
4
6
8
10
rightfoot22 24 26 28 30 32
1
2
3
4
5
6
7
8
22 24 26 28 30 32
![Page 38: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/38.jpg)
How normal is normal?
1
2
3
4
5
6
7
8
9
rightfoot16 18 20 22 24 26 28 30 32
Is the normal distribution an appropriate model for the data?• Symmetry (skew)• Bell shapeHow can we justify this?
![Page 39: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/39.jpg)
Rolling a die
1
score0 1 2 3 4 5 6 7 8
mean: 3.5sd: 1.7mean ± 1 sd:1.8 < x < 5.2≈ 56%
![Page 40: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/40.jpg)
Rolling two dice
1
2
3
4
5
6
7
score0 2 4 6 8 10 12 14
mean: 7sd: 2.4mean ± 1 sd:4.6 < x < 9.4≈ 64%
![Page 41: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/41.jpg)
How normal is normal?
1
2
3
4
5
6
7
8
9
rightfoot16 18 20 22 24 26 28 30 32
Is the normal distribution an appropriate model for the data?• Symmetry (skew)• Bell shapeHow can we justify this?
![Page 42: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/42.jpg)
Some ideas for investigations
![Page 43: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/43.jpg)
What do ‘good’ distributions look like? Experimental not sampled Not grouped (but perhaps rounded
values) Reasonable sample size Histogram with frequency rather
than relative frequency on vertical axis
Continuous? THINK – PAIR – SHARE: ideas for
experimental distributions
![Page 44: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/44.jpg)
Mark the mid-point
![Page 45: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/45.jpg)
Good shot?
![Page 46: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/46.jpg)
Are you psychic?
![Page 47: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/47.jpg)
![Page 48: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/48.jpg)
![Page 49: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/49.jpg)
![Page 50: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/50.jpg)
![Page 51: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/51.jpg)
![Page 52: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/52.jpg)
![Page 53: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/53.jpg)
![Page 54: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/54.jpg)
![Page 55: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/55.jpg)
![Page 56: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/56.jpg)
![Page 57: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/57.jpg)
![Page 58: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/58.jpg)
1. Which country has today the lowest death rate during the 1st year of life (i.e. infant mortality): Singapore, Sweden or Venezuela?
![Page 59: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/59.jpg)
2.Which country has the lowest infant mortality today: Nicaragua, Sri Lanka or Turkey?
![Page 60: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/60.jpg)
3. In which country is the average income per person highest today: Botswana, Egypt or Moldova?
![Page 61: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/61.jpg)
4. In which country do people live the longest on average today: Botswana, Egypt or Moldova?
![Page 62: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/62.jpg)
5. In which country today do women on average marry at the oldest age: Algeria, Canada or the Philippines?
![Page 63: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/63.jpg)
6. Which country has the fewest number of children per woman today: Tunisia, Bangladesh or Argentina?
![Page 64: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/64.jpg)
7.Which country emits most tones of CO2 per person today: China, France or USA?
![Page 65: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/65.jpg)
1. Which country has today the lowest death rate during the 1st year of life (i.e. infant mortality): Singapore, Sweden or Venezuela? Answer: Singapore
![Page 66: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/66.jpg)
2. Which country has the lowest infant mortality today: Nicaragua, Sri Lanka or Turkey? Answer: Sri Lanka
![Page 67: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/67.jpg)
3. In which country is the average income per person highest today: Botswana, Egypt or Moldova? Answer: Botswana
![Page 68: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/68.jpg)
4. In which country do people live the longest on average today: Botswana, Egypt or Moldova? Answer: Egypt
![Page 69: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/69.jpg)
5. In which country today do women on average marry at the oldest age: Algeria, Canada or the Philippines? Answer: Algeria
![Page 70: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/70.jpg)
6. Which country has the fewest number of children per woman today: Tunisia, Bangladesh or Argentina? Answer: Tunisia
![Page 71: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/71.jpg)
7. Which country emits most tones of CO2 per person today: China, France or USA? Answer: the USA
![Page 72: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/72.jpg)
Asking meaningful questions
Two way tables
![Page 73: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/73.jpg)
Two-Way Tables39 of the 120 students in 12MAT failed the probability practice test. As it turns out, even of the 76 students who did do regular homework, 21 students failed the test.a) Represent the data in a table.b) Write down at least one stupid question.c) Write down one question each relevant
to the teacher, a lazy student and a student with other commitments.
d) Make a case for doing homework.
![Page 74: Probability – Risk-Free](https://reader035.vdocument.in/reader035/viewer/2022062310/56815fe8550346895dceeb54/html5/thumbnails/74.jpg)
Two-Way Tables
b) Write down at least one stupid question.
c) Write down one question each relevant to the teacher, a lazy student and a student with other commitments.
d) Make a case for doing homework.
passed
failed
total
homework 55 21 76no homework
26 18 44
total 81 39 120