What is probability?

Today’s class in Bioinformatics: Models and Algorithms (a graduate/college senior course) went fairly well.

I started by collecting the first assignment and asking if there were any questions, about anything in the class.  I used a bit longer wait time than I usually use for that prompt, and was rewarded by having a reasonable question asked by someone who had clearly been hesitant to ask.  I’ve been finding that “Wait Time” is one of the most powerful techniques in Teach Like a Champion.

The first part of class was just a quick summary of DNA sequencing techniques, with an emphasis on the effect the different technologies had on the sequence data collected (read lengths, number of reads, error models).  Many of the students had already had much more extensive coverage of DNA sequencing elsewhere (there is an undergraduate course about half of which is sequencing technology), and several students were able to volunteer more up-to-date information about library preparation than I have (since they worked directly with the stuff in wet labs).

I reserved the last 15 minutes of the class for a simple question that I asked the students “What is probability?”

I managed to elicit many concepts related to probability, which I affirmed and talked about briefly, even though they weren’t directly part of the definition of probability.  This included things like “frequency”, “observable data”, and “randomness”. One volunteered concept that I put off for later was “conditional probability”—we need to get a definition of probability before we can deal with conditional probability.

Somewhat surprising this year, as that the first concept that was volunteered was that we needed an “event space”.  That is usually the hardest concept to lead students to, so I was surprised that it came up first.  It took a while to get someone to bring up the concept of “number”—that probabilities are numeric.  Someone also came up with the idea “the event space equals 1”, which I pressed them to make more precise and rigorous, which quickly became that the sum of probabilities of of events is 1.  I snuck in that probabilities of events meant a function (I usually press the students to come up with the word “function”, but we were running low on time), and got them to give me the range of the function.

Someone in the class volunteered that summation only worked for discrete event spaces and that integration was needed for continuous ones (the same person who had brought up event spaces initially—clearly someone who had paid attention in a probability class—possibly as a math major, since people who just regard probability as a tool to apply rarely remember or think about the definitions).

So by the end of the class we had a definition of probability that is good enough for this course:

  • A function from an event space to [0,1]
  • that sums (or integrates) to 1.

I did not have time to point out that this definition does not inherently have any notion of frequency, observable data, or randomness.  Those all come up in applications of probability, not in the underlying mathematical definition.  One of my main purposes in asking a simple definitional question (material that should have been coming from prerequisite courses) was to get broad student participation, setting the expectation that everyone contributes.  I think I got about 50% of the students saying something in class today, and I’ll try to get the ones who didn’t speak to say something on Monday.  Unfortunately, I only know about 2 names out of the 19 students, and it takes me forever to learn names, so I may have to resort to random cold calling from the class list.

In retrospect, I wish I had spent 5–10 minutes less time on DNA sequencing, so that there was a bit more time to go into probability, but it won’t hurt to review the definition of probability on Monday.

(This post also on my personal blog at https://gasstationwithoutpumps.wordpress.com/2015/10/02/what-is-probability/ .)


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s