Fudamentals of Probability

POLS 3220: How to Predict the Future

Warmup

Prediction Is Hard

“Stocks have reached what looks like a permanently high plateau.”

—Irving Fisher, Economist, October 1929

Prediction Is Hard

“I believe it is peace for our time.”

—Neville Chamberlain, September 30, 1938

Hitler invaded Poland less than a year later, on September 1, 1939.

Prediction Is Hard

“There’s no chance that the iPhone is going to get any significant market share. No chance.”
—Steve Ballmer, Microsoft CEO, 2007

Prediction Is Hard

Prediction Is Hard

“The overall risk to the American public does remain low.”
—CDC Director Robert Redfield, March 6, 2020

Exactly one week before the White House declared a national emergency.

Prediction Is Hard

For those keeping track at home, that’s 15 days before the official start of the invasion. The picture is of a convoy of Russian armored vehicles heading towards Ukraine.

Prediction Is Hard

• Looking back at history, it sometimes seems like events unfold in a linear, even predictable, fashion.
• We often underestimate how surprising historical events were to the people living through them.
• These bad predictions all share some combination of vagueness and neglect of uncertainty.
• Making good predictions is essential for making good decisions—in politics, public policy, business, and daily life.
• In the first few weeks of class, we’ll introduce probability, a language that helps us think about and communicate our uncertainty about outcomes.

Probability

What Is Probability?

• The language of uncertainty

• Allows us to communicate clearly and precisely how certain we are that an event will happen.

A probability is a number between 0 and 1 that expresses how likely it is that an event will happen.

• 0% means “impossible”
• 50% means “equally likely to happen or not”
• 100% means “certain”

Developing an intuition for intermediate values is trickier…

What Is Probability?

In some contexts, probability statements seem natural:

• 40% chance of rain on Tuesday
• 12% chance the Dodgers win the World Series
• 2.7% chance of rolling double sixes

In other contexts, feels weird:

“I believe there is 42% chance of peace in our time!”

-Neville Chamberlain, September 30, 1938

(My goal is to get you comfortable with such statements.)

What Is Probability?

• Many of us develop an intuition for probability by playing games of chance.

Frequentist Interpretation

• If you flip a coin 200 times, you’d expect it to come up heads roughly half the time.

• This illustrates the frequentist interpretation of probability:

  • Probability = the frequency we expect an event to occur in a large number of repeated trials.

• Formalized in the Law of Large Numbers:

  • If the probability of an event is \(p\), then the proportion of times that event occurs will approach \(p\) in a (very) large number of repeated trials.

Law of Large Numbers

The entire business model of casinos/lotteries rests on the Law of Large Numbers.

Maybe this guy comes in one night and puts $35,000 on lucky number 32. If 32 hits, the casino pays out over $1.2 million!

Law of Large Numbers

But if you get 10,000 of those guys placing bets like that, 37 lose for every 1 that wins…

Odds

Sometimes, like in casinos, it will be useful to express probability as odds.

• 50% \(\rightarrow\) 1:1 (“equally likely to happen or not”)
• 75% \(\rightarrow\) 3:1 (“three times more likely to happen than not”)
• 10% \(\rightarrow\) 1:9 (“nine times less likely to happen than not”)

To convert probability \(p\) to odds, take \(p:1-p\) and simplify.

To convert odds \(p:q\) to probability, divide \(\frac{p}{p+q}\).

Practice Problems

1. What is 80% expressed as odds?
2. What is 2:1 expressed as a probability?

What if the event can’t be repeated?

For events that can be repeated a large number of times, frequentist interpretation is pretty intuitive.

• But what about the sorts of questions I asked in the intro survey?

Subjective (Bayesian) Interpretation

Probability is not an innate characteristic of the event, but a statement about the observer’s knowledge.

Subjective (Bayesian) Interpretation

If you can precisely measure a coin’s initial velocity and rate of spin, you can perfectly predict what side it will land on (Diaconis et al. 2007).

Subjective (Bayesian) Interpretation

Probability is not an innate characteristic of the event, but a statement about the observer’s knowledge.

• Probability = “degree of belief”
• A probability statement (e.g. 80% chance of rain) communicates the forecaster’s confidence that the event will occur.
  • “Based on the information I have, I think it is four times more likely to rain than to not rain.”

Are people good at probability?

No. Assessing probability is not something that comes naturally to people.

The Equivalent Bet Test

• Here’s a useful mental trick for improving your probability calibration.

• Reframe the problem as a choice between bets.

• For example, “Would I rather have $100 if Karl Marx was born before Queen Victoria or if I draw a red ball from this jar?”

  • If you chose the first bet, it suggests you are more than 33% certain.
  • If you chose the second bet, it suggests you are less than 33% certain.

• Alter the contents of the jar until you’re indifferent between the two bets.

  • I personally prefer using a spinner wheel.

• This is called the Equivalent Bet Test (Spetzler & Staël Von Holstein, 1975; Galef, 2021)

Try it with a volunteer!

Practice

• Take 10 minutes to play “Two Truths and a Lie” with your tablemates. Pick a person to tell two truths and a lie about themselves, and everyone else must guess which is the lie.

• Our twist on the game is that everyone should write down a probability that their chosen statement is a lie. Use the Equivalent Bet Test to help calibrate your probability.

• (Feel free to play multiple rounds if there is time remaining.)