Getting estimates from a small sample

• How would you estimate how many Canadians have had Covid in the last 6 months?

• Can’t ask more than 35 million people!

• Can you get a good estimate from a much smaller group?

A simple estimate

• You ask 100 “random” people. 35% had Covid in the last 6 months.

• Is \(38.25 \times \frac{35}{100} = 13.4\ \text{Million}\) a good estimate?

• How would you be able to check?

Lets do a simple example

• Suppose we wanted to estimate how many people would vote a certain way in an election

• We’ll do a simple version of this, where we want to count how many triangles are in a random area on a page

• Think of the page as the population in a town. Too slow to ask everyone so we just take a sample of 10% (each square). 10 times the number in the square is an estimate for total number of triangles in the town

• Do with a partner, record your results

Lets see the results

Lets see the true value

Making this harder

• No we’re going to do this again on the next page

• Pick one square and repeat, tell me your answer again

The results

What this means

• It can be hard to get a sample that actually represents the population well!

• This is even harder with people, who are all very different

• Different types of people live in different areas

• If you just pick a random area, you may not get a representative sample

• This is why it is so hard to predict elections

Doing this in the real world

Who knows what this is?

• Previously, ring random numbers in the phonebook to get a random sample

• Some newer ways of doing this, like using XBox users to estimate who will win the US presidential election

Some history

• \(\pi\) is a pretty weird number with some interesting properties

• How to actually compute \(\pi\) has been studied for centuries

• Original example related to French aristocrats gambling (Buffon’s Needle)

• Will first see a very simple example, using random numbers

Throwing Darts at a Board

• Suppose you have a square board with a circle inscribed inside it

• If you throw a dart at this square, what is the probability it will land inside the circle?

• If the length of the sides of the square is 1, it has area…

• Area of Square = 1

• If the circle fits exactly inside it, it has radius and area…

• Radius = 0.5 and area \(\frac{\pi}{4}\)

• What’s the ratio of the area of the circle to the square?

Getting \(\pi\)

• So if the dart is thrown randomly at the board, the probability it ends up inside the circle is

Random Numbers

• To solve this we need to generate random darts hitting the board

• Computers actually can’t compute random things easily

• Instead compute numbers that are pseudo-random, that appear random to us

• Can you do better than a computer?

• Try get an estimate of \(\pi\) by creating random darts

Experiment

• Try to drop a pen at random on the page

• Only count those inside the square

• Count how many inside the circle, how many total

An Example

An Example

The Original Problem

The original problem is from George-Louis Leclerc, Comte de Buffon

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?

Buffon’s Needle

The probability

The probability a needle crosses one of the gray strips is

\[ P = \int_{\theta=0}^{\pi/2}\int_{x=0}^{(l/2)\cos\theta}\frac{4}{t\pi}dxd\theta = \frac{2l}{t\pi} \]

Using this

• In 1901 an Italian math teacher claimed to have done this 3400 times and estimated \(\pi\) to 6 decimal places

• You can actually use statistics to show he very likely cheated!

Random Numbers now

• Used to solve lots of difficult problems in machine learning and statistics

• The way pseudo-random numbers are generated uses math from the 1600s

• Still very hard to get truly random numbers!

How to disclose sensitive information safely

• Suppose we wanted to estimate what proportion of people here, \(p\), have evaded the skytrain/bus fare?

• You might not be comfortable answering that question!

• Is there a way to estimate the proportion of people who hop the fare gate, while keeping your data private?

A simple Rule

How to estimate p

Can we estimate p?

• You might answer Yes even if that’s not your true answer

• If you answer yes, no one will know for certain if its the truth

• We can still get an estimate of the true answer

Some quick math

\[ P(\text{Answer Yes}) = P(\text{Flipped Heads and Yes}) + P(\text{Flipped Tails}) \]

Modern Uses

• This might seem like a very simple idea, but is essentially what is known as differential privacy

• Lots of famous examples where people were identified by their anonymous data

• Every big tech company (Apple, Facebook, Google) uses differential privacy to store your data now

Wrap Up

• Statistics comes up with ways to solve real world problems involving data

• The skills in a statistics/data science/actuarial degree will always be useful

• Questions?