Section 6–1 Chance and likelihood

(Chance / Probability / Subjective aspects)

In this chapter, Dr. Sands discusses various concepts of probability that can be applied in statistical mechanics and quantum mechanics. In this section, the three interesting points discussed are a chance, probability, and its subjective aspect.

1. Chance:

“…By chance, we mean something like a guess. Why do we make guesses? We make guesses when we wish to make a judgment but have incomplete information or uncertain knowledge (Feynman et al., 1963, section 6.1 Chance and likelihood).”

Dr. Sands provides examples of chance such as radio reports of tomorrow’s weather and the likelihood of an earthquake of a certain size in a particular location in the future. Similarly, physicists might ask the question: “What is the chance that a Geiger counter will measure twenty counts of radiation in the next ten seconds?” However, chance is a non-technical word that is used daily when we are talking about likely events taking place whereas probability is a technical term that refers to a precise observation of this chance. More important, a physical theory involves some guesswork when physicists want to make a judgment but they have incomplete and uncertain knowledge.

During a Messengers Lecture, Feynman also uses the word chance when he says that “[a]n example of these tacit assumptions which I mentioned, about which we are too prejudiced to understand the real significance, is such a proposition as the following. If you calculate the chance for every possibility - say it is 50% probability this will happen, 25% that will happen, etc., it should add up to 1. We think that if you add all the alternatives you should get 100% probability. That seems reasonable, but reasonable things are where the trouble always is (Feynman, 1965, pp. 155-156).”

2. Probability:

“By the ‘probability’ of a particular outcome of an observation we mean our estimate for the most likely fraction of a number of repeated observations that will yield that particular outcome (Feynman et al., 1963, section 6.1 Chance and likelihood).”

Dr. Sands defines the probability of a particular outcome as an estimate for the most likely fraction of a number of repeated observations that will yield that particular outcome. He provides an example of tossing an “unloaded” coin in a number (N) of times. If N_A is an estimate of the most likely number of our repeated observations that will give a specified result A, say “heads,” then the probability of observing A can be represented as P(A) = N_A/N. This definition of probability requires an important condition in which the occurrence is a possible outcome of repeatable observations. Interestingly, he mentions that it is unclear whether it would make sense to consider the probability of a ghost in a house because an observation of this outcome is not repeatable.

In Théorie analytique des probabilités, Laplace (1840) writes that “[t]he probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.” This is a theoretical definition of probability that is based on the principle of indifference (or insufficient reason). On the contrary, we can introduce an empirical definition of probability that may not agree numerically with this theoretical definition of probability. For example, the empirical probability of an event is the ratio of the number of outcomes of a specified event to the total number of outcomes in an actual experiment; it is not based on a theoretical sample space.

3. Subjective aspects:

“…You may have noticed another rather ‘subjective’ aspect of our definition of probability (Feynman et al., 1963, section 6.1 Chance and likelihood).”

Dr. Sands’ explains that the concept of probability has its subjective aspects. In other words, the probability is dependent on a physicist’s knowledge and one’s ability to make estimates. For instance, if we can take a brief look at an opponent’s hand in a card game, our probability of winning can be deduced differently. In general, the number N_A is “our” best estimate of what might occur in a total number N of our imagined observations. Additionally, when we refer to the number N_A as an “estimate of the most likely number,” it does not mean we definitely expect to observe N_A, but it is possible to have other numbers near N_A. In short, probabilities are not “absolute” numbers and they may become different numerically if our knowledge changes.

Feynman might explain the subjective aspect of probability as follows: “NASA told Mr. Ullian that the probability of failure was more like 1 in 10⁵. I tried to make sense out of that number. Did you say 1 in 10⁵? That's right; 1 in 100,000. That means you could fly the shuttle every day for an average of 300 years between accidents every day, one flight, for 300 years which is obviously crazy! Yes, I know, said Mr. Ullian. I moved my number up to 1 in 1000 to answer all of NASA's claims that they were much more careful with manned flights, that the typical rocket isn’t a valid comparison, et cetera and put the destruct charges on anyway. But then a new problem came up: the Jupiter probe, Galileo, was going to use a power supply that runs on heat generated by radioactivity. If the shuttle carrying Galileo failed, radioactivity could be spread over a large area. So the argument continued: NASA kept saying 1 in 100,000 and Mr. Ullian kept saying 1 in 1000, at best (pp. 179-180).”

Questions for discussion:

1. What is the meaning of chance?

2. How would you define the concept of probability?

3. What are the subjective aspects of probability?

The moral of the lesson: Any physical theory is a kind of guesswork and probability is a system for making guesses.

References:

1. Feynman, R. P. (1988). What Do You Care What Other People Think? New York: W W Norton.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Laplace, P.-S. (1812). Théorie analytique des probabilités. Paris: Courcier.