5/28/2015

## What is randomness?

Most people have some idea what a random event is.

• Flipping a coin
• Drawing a card from a deck
• Rolling a die

The concept of randomness isn't just restricted to games. In statistics, we rely on the concept of randomness.

• To describe uncertainty
• To collect representative samples
• To come up with methods to draw conclusions and make estimations

## Defining Randomness

What does it mean for something to be random? First, we need to establish definitions

• Let's call something which has a random outcome an trial.
• We'll call the possible outcome of a trial events.

So what makes a trial random?

• For the trial, we don't know what event will occur ahead of time

Example: Flipping a coin

• The actual flipping of the coin is a trial
• The possible events are getting a heads or getting a tails

## Random $$\ne$$ Unpredictable

Random trials only have a certain number of outcomes

• If we flip a coin, it can only land heads up or tails up
• If the coin is fair, both events are equally likely

They are well-behaved in the longterm

• If we flip a coin once, we don't which side will land up
• In the longterm, however, we should see about half heads and half tails

## In Real Life

We can often view real-world events as random trials. Think about a morning commute.

• When we leave in the morning, we're can't be sure exactly how long it will take to get to work (or school)
• Traffic or construction can hold us up
• It's not completely unpredictable, however
• We might know how long an average (or expected) commute is, and can plan around it

## Using Randomness

Why do statisticians care about randomness? Mostly to eliminate bias in our studies.

• We usually need to estimate values we don't know, like the percentage of people who will vote for a presidential candidate
• We can try to estimate this by asking a fraction of the people as they leave their polls
• This only works if we get a representative cross section
• If we poll too many people from one party, we might predict that they'll win when they didn't
• We ensure that our samples are representative by making sure they are random
• Unfortunately, people are very poor at operating randomly

## Pick a Number at Random

$1\quad 2\quad 3\quad 4$

Did we get random results?

• Almost 75% of people choose 3
• 20% chooose 3 or 4
• Only 5% of people choose 1

## So what do we do?

How can statisticians and researchers get random samples if we can't trust ourselves?

• Historically, we used what are called random number tables
• Modern researchers used software
sample(1:20, 5)
## [1]  2 19  7 10  9
rnorm(n = 5, mean = 100, sd = 5)
## [1]  95.85033  96.60495  99.02627  95.46636 101.26399