Tuesday, May 13, 2014

Feynman, Community College Students and Probability

(Students frequently rise to the challenge when teachers raise the bar. Give them something to stretch their minds with and students will embrace it with vigor and purpose.
Elementary statistics is a transfer course at California’s community colleges for the CSU/UC systems. A major part of this course is probability, the workhorse of statistics. What if community college students were asked to read Richard Feynman’s lecture on probability? What would they make of it? Richard Feynman (1918-1988) won the Nobel Prize for physics in 1965 for his seminal contributions to quantum electrodynamics. He is celebrated for his physical insights and for his ability to clarify complex concepts for the general audience. His fame grew when he gave a series of lectures on physics at the California Institute of Technology for undergraduates from 1961-1963 that became the three-volume “The Feynman Lectures on Physics.” More than four decades later, the “Red Books” are still being read and still continue to inspire. The lectures are now available online. His lecture on probability challenged several community college students who found it fascinating and engrossing.)

Josh found the discussion on the uncertainty principle most interesting, Nature is probabilistic rather than deterministic, reasons enough for Josh to focus on mastering probability. Feynman says that “the ideas of probability are certainly useful in describing the behavior of the 1022 or so molecules in a sample of a gas, for it is clearly impractical even to attempt to write down the position or velocity of each molecule.” Hence his conclusion: “We now believe that the ideas of probability are essential to a description of atomic happenings,” and “our most precise description of nature must be in terms of probabilities.” Most statistics texts introduce probability through flipping coins or rolling dice that leaves students cold. For Josh, a connection between probability and nature at its most fundamental level is a compelling argument for understanding and working with probability.

Reese found the lecture interesting but hard to follow. He gets it, though, when Feynman says that probability can be used to make better guesses. Hilda agrees but found the deterministic/probabilistic contrast confusing. The random walk idea went over her head but she was pleased when Feynman acknowledged his own uncertainty “when he states that his theory can change with future knowledge.”

For Yikal, Feynman’s simple questions invoking probability were the lecture’s most memorable features.  “What is the chance of rain for today? This is basically asking, what is the probability that it will rain today? This helps us see whether we should take an umbrella or not. If the probability is too low, then umbrella won’t be necessary. Feynman’s conclusion: almost every choice we make is based on probability.” Also, “we can never be 100% certain that something will happen. And sometimes we know that something will happen but we just don’t know when it will happen. Every choice we make is based on the probability of the benefits and the chances that something good could come out of that system. For instance we are not 100% sure that we will get a good job based on our career but we go to school to be educated because there is a good chance of getting the job if we have degree.”

Kerlyn found Feynman’s focus on the connection between chance, different types of probability and nature most fascinating. She had vaguely heard about the Heisenberg uncertainty principle before but explained in the context of probability made the principle real for her. “If we try to ‘pin down’ a particle to a specific place, it will go faster. But if it is forced to go slow, it will spread out. Our most precise description of nature is in terms of probabilities.”

Kyle summarizes his understanding of Feynman’s lecture by quoting from it: “There are many different types of probability, such as independent, mutually exclusive, non-mutually exclusive, conditional probability and inverse probability. The uncertainty principle describes an inherent fuzziness that must exist in any attempt to describe nature. Our most precise description of nature must be in terms of probabilities. In the early days of the development of quantum mechanics, Einstein was quite worried about this problem. He used to shake his head and say, ‘But surely God does not throw dice in determining how electrons should go!’ He worried about that problem for a long time and he probably never really reconciled himself to the fact that this is the best description of nature that one can give. There are still one or two physicists who are working on the problem who have an intuitive conviction that it is possible somehow to describe the world in a different way and that all of this uncertainty about the way things are can be removed. No one has yet been successful.” For both Kerlyn and Kyle, this means that the last word on the subject is perhaps yet to be written, which is what makes the quest for knowledge so profoundly satisfying.

Jennifer found the connection between probability and chemistry in Feynman’s lecture compelling. She also made connection with she learned in her statistics class, that “regarding probability density, the area under the curve, known as the bell-curve, is equal to 1. Standard deviation is the variation from the mean.” To visually imagine standard deviation, Feynman illustrates the motion of a molecule. He describes an occurrence when ‘an organic compound’ is released from a bottle in a room. This organic compound then evaporates in the air, and the particles spread throughout, thus resulting in standard deviation.

For Aisa, a clear, declarative sentence like, “There are good guesses and there are bad guesses. The theory of probability is a system for making better guesses,” is as powerful an introduction to probability as anyone can think of. She finds Feynman’s ability to place probability in a unique perspective the main draw of the lecture. “It makes readers think of probability not just as a sort of math problem but something that happens in the real world. Feynman puts thinking and logic into a different realm, and that applies to his lecture on probability as well. He shows how probability is subjective. The answer may not always be what you hope for or want. Still, it is better to be probabilistic and realize that probability is a game of chances. I think this type of mind frame will help people think of probability in a different way.”

Sabrina’s understanding of probability grew when she worked through Feynman’s explanation of the binomial probability by breaking down the outcomes of flipping a coin and identifying some of the rules of the binomial model, such as, the observations must be repeatable, and the repeated observations must be equivalent. “He makes it clear that the observations are estimates of what will occur. The same reasoning can be generalized to any situations where there are different, but equally likely possible results of an observation. This of course makes perfect sense especially keeping randomness in mind. Feynman includes a fascinating graph that represents the idea that with an increase of number of tosses, the closer ‘the tendency is for the fraction of heads to approach 0.5, as compared to a smaller number of tosses where the fluctuation of deviation might be greater.’ Feynman then connects the ideas of the coin toss to random walk and motions of atoms in a gas. This is what I found most fascinating: How Feynman can take a simple concept and connect it to something like the motions of atoms in a gas. We should see more connections in our studies, whether within disciplines or between disciplines. That will motivate students far more than treating subjects as if they were disconnected from each other.”

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