## Understanding Pandemic Statistics

Real examples help to better understand how to interpret the statistics.

False Positive Paradox: A particular medical test for a disease is 96% accurate. If one has the disease, the test comes back ‘Yes’ 96% of the time, and if one does not have the disease, the test comes back ‘Yes’ 4% of the time.

If  100  of 10000 tested patients have the disease, what is the probability that the person with the diagnosis ‘Yes’ has the disease?

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Effective Average Infection Ratio: R is the effective average infection ratio for a disease, also known as the reproduction number. It is the average number of secondary infections caused by one person. (Infections caused by the secondary infections – which would be tertiary infections – are not counted). Consider 50 infected people. Suppose 49 spread the infection to nobody, but one person spreads the infection to 60 people.

What is the R-value?

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Leslie Green asks: How would you propose to deal with the COVID-19 pandemic, given information current at the time of writing (11 October 2020).

A. Severe lockdown for 4 weeks

B. Ignore it and carry on as normal

C. Partial lockdown and wait for a vaccine

D. Some other idea

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The Pfizer COVID-19 vaccine-candidate interim results from 8 Nov 2020 showed 38,955 participants in a placebo controlled double-blind trial.
94 participants  became “evaluable”, which we presume to mean they showed COVID-19 symptoms. The analysis presented was that the vaccine efficacy rate was above 90%. What is the maximum number of (genuinely) vaccinated people who showed COVID-19 symptoms?

Variability Analysis on COVID-19 Interim Trial Data by Leslie Green

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Is there any rational justification for being wary of vaccines?

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In the UK, December 2020, up to 2 million university students were potentially going home for Xmas during the middle of the COVID-19 pandemic. A new fast COVID-19 test had been developed to spot the causative virus, SARS-CoV-2. These lateral flow tests had the characteristics shown in the image. At the time, around 1% of the population had the virus within the community.

The scientific advice, given on prime-time news channels, was that a pair of negative tests meant it was safe to go home, as a negative test meant a 99.75% chance of not having the virus.

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## Online Shopper Decision

You are buying a roller on Amazon. Four similar rollers have the same functions, the same price, the same look, but different ratings.

Which product do you choose?

A. 8 positive ratings (8 total ratings)

B. 45 positive ratings (48 total ratings)

C. 90 positive ratings (98 total ratings)

D. 177 positive ratings (198 total ratings)

## Two or Three

There are 5 counters in a bag. Three are Argentinean (blue) and two are Brazilian (green). Three counters are randomly picked out of the bag, one by one. They are not returned to the bag.

What probability is higher?

A. The probability of choosing three Argentinean counters in a row.

B. The probability of choosing two Brazilian counters in a row and then one Argentinean counter.

## Secret Agent 007

Seven secret agents are on payroll of a government.

Secret agent 001 spies on a secret agent who spies on secret agent 002,

who spies on a secret agent who spies on secret agent 003,

who spies on . . . spies on secret agent 007

who spies on a secret agent who spies on secret agent 001.

Who spies on secret agent 007?

## JUICE, SODA, and LUCKY

A drinks dispenser has three labelled buttons: JUICE, SODA, and LUCKY. If you press LUCKY you are supposed to get either juice or soda with equal probability.

Each button is wired to exactly one of these functions, but each function is not necessarily wired to exactly one button.

Assuming that you are clever, but unlucky, how many button presses are required to establish exactly which button does which function?

## Motorway Problem

How to choose the shortest road that connects several towns. This is a story about local and global minimums. Excellent presentation  of soapy films.

## Unrepresentative Voting

Leslie Green explains how our voting systems can be unrepresentative.

Five different parties are standing in an election for one representative. The voting  methods:  First past the post (FPTP): the party with the highest vote-count wins.

Is the voting system representative?

The no-pets party has 30% of the total, whereas parties that want pets are the remaining 70%.  As it stands the FPTP system would vote-in the no-pets party, which does not represent the view of the electorate.  What has happened is that the pro-pet movement has splintered, dividing the vote. In order to overcome this, one can use a transferable vote scheme so people can vote for their preferred option safely, preventing the hated no-pets party from winning.

A more detailed analysis can be found in an example.

## Big Ben Maths

Big Ben is the nickname for the Great Bell of the clock at the north end of the Palace of Westminster in London, United Kingdom.

A pile of old coins helps to keep the clock mechanism accurate.  The pennies are stacked on the pendulum of the clock and have acted as weights. Adding or taking away coins effects the pendulum’s centre of mass and the rate at which it swings. A single penny will change the clock’s speed by two fifths of one second per day.

How does the weekly time change if we add 5 pennies to the pendulum?

At twelve o’clock, twelve chimes  ring from the great bell in 44 seconds. ## How long does it takes to ring the hours at 15:00 ?

## Singapore Math

Singapore math  is a teaching method based on the national mathematics curriculum used for kindergarten through sixth grade in Singapore. Singapore math, also known as the “mastery approach”: students learn a specific concept  in a linear progression before moving to the next more complex subject. Students learn and master fewer mathematical concepts at greater detail using a three-step learning process: concrete, pictorial, and abstract. These are techniques of the teaching methods: to cover fewer topics in greater depth, handling objects such as pencils, dice, or paper clips, pictorial visualisation by drawing diagrams, bar modelling (a pictorial method used to solve arithmetic problems). Bar modelling is far more efficient than the “guess-and-check” approach. A 2015 study of 140 schools in the UK by found that the mastery approach improved the speed at which students learned math skills.

This is how Jeff Bezos, the richest man on the planet teaches maths to his children.

## A Self-educated Mathematician Established the Boolean Algebra

George Boole (1815–1864)

The term “Boolean algebra” honors George Boole (1815–1864), a self-educated English mathematician. Boolean algebra captures essential properties of both set operations and logic operations. Boolean logic is credited with laying the foundations for the information age (computers and cell phones).

One of the puzzles on Boolean logic:

There are three balls X, Y and Z. They are colored red, white and blue, but not necessarily in this order. One, but only one, of the following statements is true:

X is red

Y is not red

Z is not blue

How many solutions does the puzzle have?

Think yourself before checking the solution.