How do you know you’re not in the matrix?

1. How do you know you’re not in the matrix?

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

3. Scepticism v common sense

5. Piece of paper

6. Mobius strip

7. “The mark of a civilized man is the ability to look at a column of numbers and weep”Bertrand Russell (1872- 1970)

8. 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

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

10. What’s wrong? Let us begin with an innocent statement, let a = b Multiply both sides by a to get a2 = ab Add a2 – 2ab to both sides a2 + a2 -2ab = ab + a2 -2ab This can be simplified to 2(a2 –ab) = a2 –ab Divide both sides by a2 – ab 2 = 1 ?!

11. What’s wrong? Let us begin with an innocent statement, let a = b Multiply both sides by a to get a2 = ab Add a2 – 2ab to both sides a2 + a2 -2ab = ab + a2 -2ab This can be simplified to 2(a2 –ab) = a2 –ab Divide both sides by a2 – ab 2 = 1 ?! If a = b a2 –ab = zero You can’t divide by zero!

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

13. Birthday coincidence We need to look at how many possible pairs there are. Person 1 and person 2 Person 1 and person 3 Person 1 and person 4 etc. There are 22 possible pairs with person 1

14. Birthday coincidence There are 22 possible pairs with person 1 Then person 2 with person 3 person 2 with person 4 person 2 with person 5 etc. There are 21 possible pairs with person 2

15. Birthday coincidence There are 22 possible pairs with person 1 There 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.

16. 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%)

17. 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!

18. 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?

19. 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

20. The students point of view There 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.

21. 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!

22. A Truel A 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?

23. 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

24. 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.

25. Another option? Is there a third option?

26. 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.

27. 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.

28. 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?

29. Oslo to Kristiansand For 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)

30. Not always obvious is it?

31. Ockham’s razor • Given two explanations that could be true, always believe the simplest explanation.