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E03: How 2 rig an election

Podcast

Bia and Zoey discuss some of the key mathematical concepts in voting, focusing on political elections in some Western countries, as well as Brexit.

Introduction
0:15 – Introduction on the voting system in the UK, with an example
4:27 – Condorcet’s paradox 6:00 – The French system
6:44 – The Australian system – Preferential/Alternative voting
7:42 – What defines a good voting system?
9:43 – How do we balance a good voting system with one which everyone understands
Arrow’s Impossibility Theorem & Instagram poll
11:40 – Arrow’s Impossibility Theorem
13:46 – Independent voting systems where Arrow’s theorem doesn’t apply.
15:00 – Instagram poll discussion: Tactical voting vs Voting for who you want
17:31 – Protest voting vs voting for who you want
20:20 – When it would be worth strategically voting “mathematically”
Brexit21:22 – Zoey exposing Bia as a “Remoaner”
22:50 – How Bia think the referendum should have been done. General discussion
24:05 – Have you ever not voted?
26:12 – Should 16 year olds be allowed to vote? 2
27:29 – Accessibility in a voting system
The future of voting
27:54 – The future of voting
28:52 – The issues with a voting system which takes too long (NP-hard/ NP-complete)
30:00 – Dodgson’s voting method (Lewis Carroll = Charles Dodgson)
32:52 – Final thoughts

80% of voters are strategic: “Counting Votes Right: Strategic Voters versus Strategic Parties, Filippo Mezzanotti and Giovanni Reggiani” http://economics.mit.edu/files/11177

Black History Month 2020

History of Maths

Here in UK, October is Black History Month. To celebrate, we posted a three-part mini series on our Instagram.

1. Mathematics in Timbuktu

Part one dives into the Mathematics in Timbuktu. The recently rediscovered Timbuktu manuscripts (dating back to the 13th century) demonstrate the continuous knowledge of advanced mathematics and science in Africa well before colonisation. They understood trigonometry, they created algorithms and they were advanced in astronomy!

2. Black Mathematicians

For part two of the mini series, we mention a few notable Black Mathematicians.

  • Thomas Fuller (1710-1790) was an African-born slave in America known as the Virginia Calculator for his extraordinary powers in arithmetic.
  • Marta Euphemia Lofton Haynes (1850-1980) is the first Black woman in the US to earn a PhD in Mathematics in 1943.
  • Jesse Ernest Wilkins Jr. (1923-2011) entered the University of Chicago at 13 years old and became the youngest ever student at that university.
  • Katherine Adebola Okikiolu (1965-present) is the first Black person to receive a Sloan Research Fellowship. (43 fellows have won a Nobel Prize, and 16 have won the Fields Medal in Mathematics)

3. Ancient African Mathematics

For our third and final post, we discuss the first mathematical instruments known to us – the origin of these fascinating ancient relics is African!

  • The Ishango Bone 🦴 (2,000 BC): This bone has 3 columns with different markings on them in clusters.
  • The Lebombo Bone🦴 (30,000 BC): This bone has 29 markings on it believed to be linked to a lunar cycle.

The purpose behind the creation of these relics is still disputed. According to certain historians, (e.g. Museum of Natural Sciences, Brussels, where the Ishango bone is presented), it is believed that the people of Ishango knew about prime numbers. However, some mathematicians say that the presence of prime numbers on the bone is a coincidence and that they were actually using the bone as a counting tool. Either way, it is fascinating that people over 35,000 years ago were using maths in some way – possibly before the invention of numbers!

Both of these bones were made on baboon fibulas. The baboon was a symbol of the Ancient gods of the moon which suggests an ancient connection between baboons, the moon, time, and maths.

Did you enjoy this? Make sure you’re following us on instagram @how2robabank!

E02: How 2 predict grades badly…

Podcast

In this episode, we discuss some of the mistakes that Ofqual made in their algorithm, how using “complicated” maths is not necessarily better, and share some anecdotes of their experiences with teachers and dealing with (un)conscious bias.

Timestamps
00:20 – Introduction
01:54 – Initial thoughts
02:42 – Mistake #1 – Their approach
04:43 – Mistake #2 – Data leakage
05:15 – Mistake #3 – Emphasis on the rank
06:57 – Mistake #4 – Ignoring outliers
08:31 – Mistake # 5 – No peer review
09:16 – Mistake #6 – Too precise
11:14 – Mistake #7 –Disregarded unconscious bias.
12:53 – Mistake #8: Education system in the UK.
13:30 – Ofqual considered edge cases – (almost a positive thing!)
15:00 – How we might have handled this situation
17:39 – Another example of algorithmic bias – Accounting system the Post Office used.
18:53 – Challenge: “Prison Break”. This based on “Liar’s paradox” attributed to Epimenides (amongst many other philosophers). For more challenges, presented in a more visual manner, check out our Instagram.
25:52 – Anecdotes of experiencing bias from teachers.

Useful links:
Ofqual’s reportBristol University’s study on unconscious bias 
http://www.bris.ac.uk/media-library/sites/cmpo/migrated/documents/wp221.pdfTom SF Haines’ post (Lecturer in Machine Learning at Bath University) – http://thaines.com/post/alevels2020

Challenge #3 – The Devil’s Tail – Solution

Challenges, Solution

This is the solution to Challenge #3 – The Devil’s Tail, make sure you check that post out before looking at the solution!

This Challenge is a modification of a famous problem which confused many people – The Boy-Girl paradox.

So, it turns out that “Heads” is the most probable at 67%. But why?

The devil could have any of the 3 combinations: {HT, TH, TT}, and 2 of those include a Heads! A common answer is “Equally Likely” as it is easy to assume that the devil showed us a specific coin, meaning the remaining coin is equally likely to be Heads or Tails, but this is not the case as we do not know which coin the devil showed us. Some people’s intuition may also have lead them to think it might be Tails as we have already had a Tails, but this is also not true. Hopefully this problem illustrates how probability is very deceptive and counter-intuitive!

Challenge #2 – The Elements – Solution

Challenges, Solution

This is the solution to Challenge #2 – The Elements, make sure you check out the challenge before looking at the solution!

Trust the devil. Why?

The first clue given in the challenge is the note left behind by Doctor Aro.

The key piece of information on the note says that we have to reflect on the odd things in life.

Reflecting the position of the digits in the odd numbers we get:
71 6 53 9 68
(Reflecting the position of the digits for 9 leaves it the same)

But how do we decipher the numbers?

Dr Aro was ‘dabbling in the chemical arts’ and this challenge is called ‘The Elements’, written in a similar way to the elements in the periodic table.

Using the periodic table we get:
Lu-C-I-F-Er

Lucifer i.e. the devil 😈

Were you able to save Doctor Aro?

Challenge #2 – The Elements

Challenge, Challenges

Can you save Doctor Aro?



THE CHALLENGE
Now that you and your accomplice, Ez have escaped, you have one more friend to free, known as “Doctor Aro” 🧑‍🔬 before you can proceed.

Doctor Aro had a near-death experience from her dabbling in the chemical arts. 💀 As a result, she is stuck at the gates of heaven waiting for a decision to be made.

Guarding the gates are the angel and the devil. “I can help you,” they both offer. Since you’re a badass thief, you’re not sure who to trust…

Suddenly in a chemical burst, a piece of paper appears in Doctor Aro’s handwriting pleading for your help:

“Sometimes, you have to reflect on the odd things in life. 17 6 35 9 68”

Who should you trust?
The angel 😇
or
The devil 😈?

TIP
You may need google/pen and paper.

SOLUTION
The solution to this challenge can be found at Challenge #2 – The Elements – Solution.

Challenge #1 – Prison Break – Solution

Challenges, Solution

This is the solution to Challenge #1 – Prison Break – make sure you check that post out if you haven’t already!

It turns out that Guard C is your friend!

We are given the information that each robot guard always lies for the whole day, or always tells the truth for the whole day. This information does not indicate that the guards have truth/lie days in any particular order.The robot guards could have truth/lie days as follows: Truth, Lie, Truth, Lie, …
But equally could have days as below:
Truth, Truth, Truth, Lie, …

There is no reason to presume that the guards all follow the same orders, so we can assume that each guard is independent of the others. This means a guard’s truth/lie days can be different to the other guards.

So why is Guard C our friend? Guard C said “Today, I lie.”

  • If Guard C is telling the truth, then it means Guard C is lying as indicated from the statement.
  • – If Guard C is lying, then Guard C has just told us the truth about lying.

Guard C’s statement is a paradox!

Paradoxes like the one presented in this challenge, also known as “Liar’s paradox”, are attributed to the ancient Greek seer Epimenides (fl. c. 6th century BCE). Epimenides, an inhabitant of Crete, famously declared that “All Cretans are liars”.

This paradox will arise whenever the statement refers to whoever is claiming the lie themself.

Consider: “This sentence is a lie”.

This circular logic is important in part because it creates severe difficulties for logically rigorous theories of truth; it was not adequately addressed (which is not to say solved) until the 20th century.

Challenge #1 – Prison Break

Challenge, Challenges

Can you help your friend escape? 🔒

THE CHALLENGE

Before you can begin the heist, you need to build a team of people you can trust. You received news that your good friend and former accomplice has almost escaped prison and is impersonating a guarding robot waiting for a chance to exit.

When you arrive at the prison gates, you notice there are 5 guards but cannot tell which is your friend. You only know one thing:

Each robot guard always lies for the whole day, or always tells the truth for the whole day (but your friend is not bound by these rules). Can you figure out which guard is your friend and help them escape?

  • Guard A: “Today, I tell the truth”
  • Guard B: “Yesterday, I lied”
  • Guard C: “Today, I lie”
  • Guard D: “Yesterday, I told the truth”
  • Guard E: “Tomorrow, I shall tell the truth”

TIP
This week we’ve spoken a lot about making assumptions that could lead to incorrect conclusions, so if you’re struggling, have a think about anything you may have assumed implicitly…

DISCLAIMER
The difficulty level reflects how long we might expect for someone to spend on this vs some of our future problems. Of course, some may rate this as a 3 or 4, whereas others may rate it at a 1. How difficult you find this is very relative, but the problem does not assume any prior knowledge of maths.

SOLUTION
The solution is revealed in the post Challenge #1 – Prison Break – Solution.

E01: How 2 make a decision ft. The Monty Hall Problem

Podcast

What makes a good decision? Do you think decisions are good if they were good enough at the time, or does the outcome shape your idea?

Known for its controversy, The Monty Hall Problem was popularised through a newspaper column called Ask Marilyn. In this episode, we discuss how probability can help us make a decision in The Monty Hall Problem as well as more generally. We also try to define the “wrong decision” and the circumstances under which we might regret our choices. Does the outcome of your decision imply how good it was in the first place? Later on, we briefly venture into numbers and how our minds don’t always perceive things correctly. Finally, we discuss The Two- Envelope Paradox and how the assumptions we make can lead us to the wrong conclusions.

Useful links to understanding The Monty Hall Problem: V Sauce’s video on the Monty Hall Problem. An article to help explain.

The Envelope Problem explained:https://plus.maths.org/content/two-envelopes-problem-resolutionhttps://brilliant.org/wiki/two-envelope-paradox/

This episode was recorded on 16th June 2020.