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Year 10 Probability

Three-week unit for the Enhanced Maths Year 10s at Mater Christi

Louise Hartley

on 18 September 2013

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Transcript of Year 10 Probability

{ Probability
Textbook Section 12a
Flipchart pages 1-7
or watch this...
Mutually Exclusive and Complementary Events
This is covered by the textbook Section 12b and Flipchart p8-16.
Subjective Probability - not really maths at all
See the textbook p420-422.
Do we care about Probability? Is it important?
If you have time - watch this - it will open your mind
It's a fast-paced presentation, but if you want to get an idea of why and how statistics affect your life EVERY DAY, then try and stick with it, even if some of it goes over your head it's worth it!
<---My hero, Ben Goldacre
...and this (for Venn diagrams)...
Worksheet 1.
Textbook Exercise 12a (p392 - 396).
All resources for this unit can be downloaded from https://drive.google.com/folderview?id=0BxW9xi_0IWyyOHBER3VJTEZNbGs&usp=sharing
A lot of the videos are taken from the Khan Academy and there's a whole section on the Khan Academy website that's useful for this unit. It's at https://www.khanacademy.org/math/probability/independent-dependent-probability
Our Goal
Complementary and mutually exclusive events
Two-way tables and treediagrams

The importance of probability


What's the chance????
Dependent events
Conditional Probability
Karnaugh Maps
Tree diagrams - basic

Two-way tables and tree diagrams
This is covered in the textbook, section 12c.
Textbook Exercise 12c,
What's the chance of having three sets of twins in the same family? http://www.bbc.co.uk/news/magazine-22813345
Confusing independent and dependent events can have disastrous consequences - read the sad case of Sally Clarke, wrongly convicted of triple murder in the excerpt from Bad Science "Locking you up, the ecological fallacy'.
Real-life examples of why its so important to understand dependence and independence
These misconceptions crop up in media reports all the time, see if you can find any.
NEWS FLASH!! Taller children do better at maths!
Researchers randomly selected a sample of 100 high-school children and gave them an identical maths test. The taller children on average scored more highly than the shorter children, proving that tall children are better at maths. Right?
Example of dependent events
Have a look at Worksheet 2 and for each of the 6 problems first identify whether you are dealing with independent or dependent events
From the textbook, Exercise 12d (p415-417), questions 1,2,3,6,7,8,12,13,14

These are like a combination of two-way tables and truth tables, but can represent up to three events and their complements.

This isn't in the text book. See instead the flipchart, pages 19 to 21.

The youtubes are a bit confusing, so we'll go through it in class.
We're going to do an activity in class for this one. But if you want to prep yourself, see the textbook, section 12e (p417-418) and/or the flipchart p28-p32.
Have a look at http://www.livescience.com/32767-what-are-false-positives-and-false-negatives.html.
Textbook exercise 12e, p419-420.
False positives and false negatives
If you want to dig deeper there is history and mathematical background at http://en.wikipedia.org/wiki/Bayes'_theorem
By the end of this unit you should understand and be able to solve maths problems involving:
Complementary and mutually exclusive events;
Two-ways tables and tree diagrams;
Independent and dependent events;
Karnaugh Maps;
Conditional Probability;
Subjective Probability.
Some real-world
probability applications
Textbook exercise 12b, p400-403, questions 2,3,4,11,16,17,18,20,21,23,24
The Addition Rule for Probability
Mutually exclusive
events cannot both occur simultaneously
, if you select one person at random, there's a chance of them being born on a Tuesday or Wednesday, but not both.
Complementary events
are mutually exclusive events which together make up the universal set, e.g. a person was born on a Tuesday, or they weren't born on a Tuesday.

Independent events
In a group of five people what is the probability that two or more of them are born in the same month?
Does the month that one person is born in affect the month that another person is born in?
The Multiplication Rule for Probability
*simultaneous in this context does not necessarily mean "at the same time", it means "on a given trial' - we can talk about this in class.
*unless of course we are observing groups of twins, triplets, etc.
So far we've been talking about "simultaneous" events, what happens if we have a series of events, turns, draws or observations. We can use.....
Can only represent two events
Can represent any number of events
The Monty Hall Solution
And that's it!
What's the probability of being born on the same day?
You should expect to get here by the end of lesson 1 plus homework time.
This video is at https://www.khanacademy.org/math/probability/independent-dependent-probability/basic_probability/v/basic-probability
This video is at https://www.khanacademy.org/math/probability/independent-dependent-probability/addition_rule_probability/v/probability-with-playing-cards-and-venn-diagrams
Try this activity: http://www.regentsprep.org/regents/math/algebra/apr6/pracmut.htm
This video is at https://www.khanacademy.org/math/probability/independent-dependent-probability/addition_rule_probability/v/addition-rule-for-probability
This video is at https://www.khanacademy.org/math/probability/independent-dependent-probability/dependent_probability/v/independent-events-1
This video is at https://www.khanacademy.org/math/probability/probability-and-combinatorics-topic/probability_combinatorics/v/birthday-probability-problem
This video is at https://www.khanacademy.org/math/probability/independent-dependent-probability/dependent_probability/v/monty-hall-problem
Have a go at the two-way table activity at https://www.khanacademy.org/math/probability/independent-dependent-probability/basic_probability/e/dice_probability
for you
This video is at https://www.khanacademy.org/math/probability/independent-dependent-probability/independent_events/v/events-and-outcomes-2
Two-way tables example
Exercise 12d, 1,2,3,6,7,8,12,13,14
Digression (or possibly the most interesting thing in the whole unit)
While I was putting together the section on conditional probability, I started getting really interested in screening tests for everything from HIV to drunk drivers to pregnancy to terrorists --- measuring how accurate these tests are rests entirely on understanding conditional probability.

Nothing in this frame is required knowledge for this unit, but I think it's fascinating and I wanted to share with you. If you're interested, read on...
False positives
All tests have a certain rate of
false positives
. This is defined as the probability that the test will give a
positive detection
when the thing being tested for is
not present

Mathematically, this is defined as P(positive | characteristic not present). Can you see why?
False negatives
Similarly, all tests have a certain rate of
false negatives
. This is defined as the probability that the test will give a
negative detection
when the thing being tested for
is present

Mathematically, this is defined as P(negative | characteristic present). Can you see why?
The Sensitivity of a test is the probability that a it will give a
positive result
when the characteristic is present

Mathematically, this is defined as P(positive | characteristic present). Can you see why?
The Specificity of a test is the probability that a it will give a
negative result
when the characteristic i
s not present

Mathematically, this is defined as P(negative | characteristic not present). Can you see why?

Tree diagrams - more advanced
Visualisation of conditional probability using mammograms and breast cancer.
Good overall guide to conditional
A good overview is at http://www.mathsisfun.com/data/probability-events-conditional.html
Thank you for teaching me...
technology WILL fail - always have a backup plan;
stop talking so fast;
next time, learn everyone's names at the beginning;

Full transcript