Fall 2021
Monday, 30 August 2021
goldenberg@biology.utah.edu
Announcements
First Quiz: Friday, 3 Sept. 25 min, second half of class.
First problem set Posted on Canvas; due Tuesday, 7 Sept. Via Gradescope
TA Discussion Sessions: • Mondays, 2:00 – 3:00 PM, Zoom • Fridays, noon – 1:00 PM (except this Friday), in-person
Office hours(Zoom): • Tuesdays: 9:30 - 10:30 AM • Wednesdays: 2:30 - 3:30 PM • Other times by appointment.
Send me an e-mail message!
Robert Brown
Scottish botanist, explorer of Australia and exceptional microscopist.
In 1827, observed random motions of small (1—m) particles within pollen grains.
Are they alive?
Brownian Motion Movie
E = mc2
Photoelectric effect
Brownian motion
Einstein and Maric Worked closely while both were university students (1896–1900), and after.
Married in 1902; divorced in 1919.
Some have argued that Maric made important contributions to the 1905 papers, especially special relativity.
Very little historical record.
Esterson, A., Cassidy, D. C. & Sime, R. L. (2019). Einstein’s Wife: The Real Story of Mileva Einstein-Maric. MIT Press. https://mitpress.mit.edu/books/einsteins-wife
Simulation of Brownian Motion
Motion of particles is caused by fluctuations in the random motions of solvent molecules.
Einstein made this model quantitative, allowing it to be rigorously tested.
Conclusive evidence that liquids and gasses were made up of discrete molecules.
A detailed, realistic molecular simulation is very difficult!
Heads - East Tails - West
Results of the Coin Toss Experiment
T H T T T T H T H H T T H H T H T T T H 8
T T T H H H T T H H H H H T H H T H H T 12
H H T H T T T H H T H T T H H H H T T H 11
T H T T T T T H H T T H T T H H T H H T 8
T T T H H T T T H H H H H H T H T H T T 10
H T H H H T T H T H H T H H T T T T H T 10
T H H H T H T T T H T H H H H T H T T T 10
T H T H T H T H T H T H H T T T H T T T 8
H T H T T H H T H H H T H T H T H T T H 11
H T H H T T T T H T H H T H T T T T H T 8
H T H H T T H T H T H H T T H T H T H T 10
H H T T T T H H T H H H T T T H T H T T 9
Results of the Coin Toss Experiment
H H H H T T H H T H T T T T H H H H H T 12
T T H T T T T T H T T H T H T T T T H H 6
T T H H H T H T T H T H H H T T T T H H 10
H H H T T H T T T T T T T H H H T T H H 9
T H T T H T T T T T H H T H H T T T T T 6
H T T T H H H H H T T T H T H H T H T H 11
T H H H T T T T H T T H H H H H H H H T 12
T H T H T T H T T T H H H H T H T H T T 9
H T H T H T T H H T T H H T H H T H T T 10
H T T T H H T H T H H H T T T H T H T T 9
T H H T T T T H T H H T T T T T T H H H 8
T T T T T T T H T H H T H T T T H H T T 6
Some Statistics from the Coin Toss Experiment
Total sequences: 26
Heads - East Tails - West
Total distance traveled: Pedometer distance. The number of steps times the length of each step.
The final displacement from the starting point.
Displacement from the Starting Point for a One-Dimensional Random Walk
Heads - rightTails - left
Take n random steps to the right or left.
nH = no. of heads nT = no. of tails
Final position is x . x = nH − nT
(in units of step lengths)
Generally expect a distribution of x if the random walk is repeated a large number of times.
Distribution of Displacements from Our Coin Tosses
What patterns can we find?
We need more data!
Simulation of a Random Walk: The Galton Probability Machine
Sir Francis Galton (1822-1911) • Cousin of Charles Darwin. • Attempted to find a mathematical description of genetic variation and
evolution. • Early advocate of eugenics (invented the term); improvement of humans by
selective breeding.
Clicker Question #1
Steve is very shy and withdrawn, invariably helpful but with very little interest in people or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.
Is Steve more likely to be a librarian or a farmer? A) Librarian
B) Farmer All answers count for now!
Key information: There are about 15-times as many farmers than librarians in the US. From the Bureau of Labor Statistics Occupational Outlook Handbook: https://www.bls.gov/ooh
Thinking Fast and Slow
Kahnemann is a psychologist who won the Nobel Prize in Economics for research on how people make decisions.
With his collaborator, Amos Tversky, Kahnemann determined that people respond to questions or problems in two distinct ways: • Fast, instinctual way that is often
right, but sometimes very wrong. • Slow, deliberate analysis.
Highly recommended book!
The Fast Way of Answering the Question
Steve has the characteristics I associate with a librarian: Shy, with a need for order and structure.
It makes more sense that Steve would be a librarian than a farmer.
The Slow, Probabilistic Way of Answering the Question
Steve is one of a large number of people in a population who are either librarians or farmers.
Represent farmers and librarians as marbles in a box.
All farmers (blue) and librarians (yellow) Shy, organized farmers and librarians
If we pick a marble at random, are we more likely to pick a blue marble (farmer) or a yellow one (librarian)?
Probability: Some Definitions
Outcomes – Possible results of a probabilistic process • For a coin toss: coin lands heads-up (H) or tails-up (T ) • For a roll of a six-sided die: The number of spots on the side that lands up
(1, 2, 3, 4, 5 and 6) • Distinguished from “events”, to be defined shortly
Probability – A number with a possible value from 0 to 1, associated with a single outcome. • p = 0: Outcome will never occur. • p = 1: Outcome will always occur. • The sum of the probabilities of all possible outcomes of an experiment must
equal 1. • What do we mean by this? What is implied? How do we interpret
probabilities that are greater than 0 and less than 1?
Two Interpretations of Probabilities
1. The frequentist interpretation • If the same experiment is repeated a large number, N, times, an outcome
with probability p will occur approximately N · p times. • “Law of large numbers” • Value of probabilities are defined by properties of the experiment.
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