How do you know you’re not

in the matrix?

• Sceptics claim we don’t know ANYTHING about the world around us

Scepticism

Scepticism v common sense

Piece of paper

Mobius strip

“The mark of a civilized man is the ability to look at a column of numbers and

weep”Bertrand Russell (1872- 1970)

Mathematics

“Math – that most logical of sciences – shows us that the truth can be highly counterintuitive and that sense is hardly common”

K.C. Cole

Problems

Each group will be given a mathematical problem. You have 4 minutes to figure out the answer.

What’s wrong?Let us begin with an innocent statement, let

a = bMultiply both sides by a to get

a2 = abAdd a2 – 2ab to both sidesa2 + a2 -2ab = ab + a2 -2abThis can be simplified to

2(a2 –ab) = a2 –abDivide both sides by a2 – ab

2 = 1?!

What’s wrong?Let us begin with an innocent statement, let

a = bMultiply both sides by a to get

a2 = abAdd a2 – 2ab to both sidesa2 + a2 -2ab = ab + a2 -2abThis can be simplified to

2(a2 –ab) = a2 –abDivide both sides by a2 – ab

2 = 1?!

If a = b

a2 –ab = zero

You can’t divide by zero!

Birthday coincidence

What is the chance/probability that two people in a group of 23 randomly selected people have the same birthday?

Birthday coincidence

We need to look at how many possible pairs there are.

Person 1 and person 2Person 1 and person 3Person 1 and person 4 etc.

There are 22 possible pairs with person 1

Birthday coincidence

There are 22 possible pairs with person 1

Then person 2 with person 3person 2 with person 4person 2 with person 5 etc.

There are 21 possible pairs with person 2

Birthday coincidence

There are 22 possible pairs with person 1There are 21 possible pairs with person 2

It follows there are 20 possible pairs with person 3, 19 with person 4, 18 with person 5 etc.etc.

Birthday coincidence

The total number of different pairs of people is therefore;

22 + 21 + 20………….+ 3 + 2 + 1 = 253

Since there are 365 possible birthdays (we’ll ignore 29th February for simplicity!), the chance of two people having the same birthday is 253/365 = 0.69 (69%)

Birthday coincidence

What is the chance/probability that two people in a group of 23 randomly selected people have the same birthday?

69%

If there are over 30 people the chance is over 100%, worth a bet at a party!

Class size

There are three classes in year 3. One class contains 12 students, one class contains 25 students, and one class contains 23 students. What is the average class size from

• the teachers’ point of view?• the students’ point of view?

The teacher’s point of view

The teacher has to teach 3 classes, one of 12 students, one of 25 students, and one of 23 students.

Average class size = (12 + 25 + 23)/3 = 20

The students point of viewThere are 12 students in a class of 12. When asked the size of their class they will all say 12

12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12.

25 students will say 25!

25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25.

23 students will say 23!

23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23.

The students point of view

Average class size for a student =

((12 x 12) + (25 x 25) + (23 x 23))/(12 + 25 + 23)

= 21.6

(Remember the average from the teacher’s point of view was 20!

A TruelA truel is similar to a duel, except there are three participants rather than two.

One morning Mr Black, Mr Grey and Mr White decide to resolve a conflict by truelling with pistols until only one of them survives.

Mr Black is the worst shot, he hits the target only 33% of the time.Mr Grey is a better shot, hitting the target 66% of the time.Mr White is the best shot, he hits the target every time (100%)

To make the truel fairer, Mr Black shoots first, followed by Mr Grey (if he is still alive!), then Mr White (if he is still alive). They go round again until only one is left alive.

Where should Mr Black aim his first shot to give him the best chance to survive?

Option 1

If Mr Black shoots at Mr Grey and hits him Mr White will then shoot Mr Black and Mr Black will die because Mr White never misses.

I

Option 2

If Mr Black shoots at Mr White and hits him the next shot will be taken by Mr Grey. Mr Grey is only 66% accurate so there is a chance Mr Black may survive to fire back at Mr Grey and win the Truel.

This would appear to be a better option then option 1.

Another option?

Is there a third option?

Option 3

Mr Black could aim in the air. Mr Grey has the next shot and will aim at Mr White because he is more dangerous. If Mr White survives he will aim at Mr Grey who is a more dangerous opponent.

By aiming in the air, Mr Black is allowing Mr Grey to eliminate Mr White or vica versa.

The best strategy

Option 3 is the best strategy. Eventually Mr Grey or Mr White will eliminate each other, allowing Mr Black the first shot in a duel instead of a truel.

Oslo to Kristiansand

A student drives from Oslo to Kristiansand (300 km) and back again. Her average speed for the first half of the journey is 50 km/h (Oslo to Kristiansand) and her average speed back is 20 km/h. What is her average speed for the entire journey?

Oslo to KristiansandFor Olso to Kristiansand, time = distance ÷ speed = 300 ÷ 50 = 6 hours

For Kristiansand/Oslo, time = distance ÷ speed = 300 ÷ 20 = 15 hours

Average speed = distance ÷ time = (300 + 300) ÷ (6 + 15) = 28.6 km/h.

(not (50 + 20)/2 = 35 km/h)

Not always obvious is it?

Ockham’s razor

• Given two explanations that could be true, always believe the simplest explanation.