3.1.Probability

Acknowledgments: many examples taken from the book Intuitive Biostatistics,4th edition

Bárbara D. Bitarello

2025-10-01

Outline

  • Understand once and for all that statistics and probability are NOT intuitive

  • Definitions, terminology & notations

  • Next week:

    • Foundational concepts: random trials, exclusive and non-exclusive outcomes, probability rules

    • Probability distributions and sampling distributions

Motivation

Why sample?

We almost never sample an entire population. So we can’t nail down parameters from populations.

But we can make educated guesses of parameters by making estimates from samples.

Estimation with uncertainty

  • We know that estimates will differ from parameters by chance.

  • So, we must incorporate uncertainty in our estimation.

  • How can we rigorously quantify this uncertainty?

Statistics (and probability, in particular) are not intuitive

“If something has a 50% chance of happening, then 9 times out of 10 it will” — Yogi Berra

Meaning of “intuitive”

  1. “easy to understand”

  2. “instinctive, or acting on what one feels to be true even without reason”

Each row is a basketball player

Do you spot any patterns?

Generalizing from a sample to a population seems to be hard-wired into our brains

Images: AFP/Getty Images/Warner Bros Records. Taken from: https://www.bbc.com/future/article/20140730-why-do-we-see-faces-in-objects

Statistics (and probability, in particular) are not intuitive

  • we see patterns in random data

Pick a range that you think has a 90% chance of containing the right answer

Don’t consult anything/anyone. The goal is to quantify your uncertainty. If you truly have no idea, you can use the broadest range that is plausible (which you can be 100% sure includes the answer) but try to narrow down your answers to a range that you are 90% sure contains the right answer.

  1. Martin Luther King Jr’s age at death
  2. Length of the Nile river (miles; km)
  3. Number of countries in OPEC1
  4. Number of books in the Old Testament
  5. Diameter of the moon (miles; km)
  6. Weight of an empty Boeing 747 (lbs; kg)
  1. Year Mozart was born
  2. Gestation period of an Asian elephant (days)
  3. Distance from London to Tokyo (miles; km)
  4. Deepest know point in the ocean (miles; km)

99% of people are overconfident in this test

These questions were used in a study by Russo and Shoemaker (1989) and 99% of \(~1,000\) participants, but most of them included narrow ranges that included 30 to 60% of correct answers.

Similar studies done with experts answering questions in their own field of expertise led to similar results.

Statistics (and probability, in particular) are not intuitive

  • we see patterns in random data
  • we tend to be overconfident

That’s all for today

Forrest says "And that's all I wanted to say about that"

From: makeameme.org