**{ Probability**

Review

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?

http://www.sbs.com.au/news/article/1708501/What-are-your-chances-of-winning-Oz-Lotto

PROBABILITY

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)...

Exercises

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

**Review**

**Complementary and mutually exclusive events**

**Two-way tables and treediagrams**

**Independent**

events

events

**The importance of probability**

**Conditional**

probabilty

probabilty

**Karnaugh**

maps

maps

**Recap**

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,

1

,

2

,

3

,

4

,

5

,

7

,

8

,

10

,

13

,

16

,

17

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

Exercises

https://www.khanacademy.org/math/probability/independent-dependent-probability/dependent_probability/e/dependent_probability

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

Exercises

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.

Exercises

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

Exercises

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.

**Dependent**

events

events

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?

no*

The Multiplication Rule for Probability

Watch!

*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

https://www.khanacademy.org/math/probability/independent-dependent-probability/independent_events/v/compound-probability-of-independent-events

This video is at https://www.khanacademy.org/math/probability/independent-dependent-probability/dependent_probability/v/monty-hall-problem

THINK "NOT"

Have a go at the two-way table activity at https://www.khanacademy.org/math/probability/independent-dependent-probability/basic_probability/e/dice_probability

Activity

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

Exercises

https://www.khanacademy.org/math/probability/independent-dependent-probability/independent_events/e/independent_probability

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?

Sensitivity

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?

Specificity

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

probability.

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;