I recently lead an intense course for incoming undergraduates who are going to concentrate in the sciences. I started the class by talking about sets, then moved into counting problems, followed by calculating probabilities, independence/dependence, unions & intersections, probability distributions, and finally hypothesis testing. It was a bit of a whirlwind! I had a lot of goals for myself for this class, which I won’t list out here at this time, but one of the goals was to balance the really serious public health examples (having TB and HIV, time to death, etc) with some light-heartedness.
For example, in the final exam I tried to include a little math/stats humor:
Blake needs to do each of these things today:
- Renew subscription to Statistician’s Fashions Quarterly
- Finish writing the love-song titled You are the Only [n choose 0] For Me
- Apply to compete on America’s Next Top Statistical Model
Assuming that Blake could only do one activity at a time, how many different ways could Blake order these activities?
Every day of the class, the students were asked to comment on which topics were the most difficult or easy to understand in an online discussion (sprinkled with pertinent Biostatistics Ryan Gosling entries) on the course website. At the beginning of every class we had a few warm-up exercises that dealt with the topics the students mentioned in the discussion. I also tried to keep these warm-ups pretty light.
Here is an example:
Archibald has packed his suitcase to go on a trip with the following items:
- Pants: 1 pair of red pants, 1 pair of blue jeans, 1 pair of green shorts
- Socks: 1 pair of orange socks, 1 pair of argyle socks, one pair of ankle socks, one pair of black socks
- Shoes: 1 pair of sneakers, 1 pair of dress shoes
- Shirts: 1 purple T-shirt, 1 yellow T-shirt, 1 plaid shirt, 1 pink tank top
If an outfit always consists of 1 shirt, 1 pair of shoes, 1 pair of pants, and 1 pair of socks, how many different outfit possibilities does Archibald have?
As we began to discuss how to approach this problem, we talked about the total number of combinations, and how some combinations of garments could look good while others would not be as easy on the eyes.
Thinking back on this, a good follow up question would have been to ask each of them to define the set of good outfits and the set of outfits where the clothes didn’t go well together. Then I could ask them to find the probability of choosing a good outfit if each outfit could be chosen with equal probability.
I started poking around in combinatorics textbooks and I found that “figuring out what to wear” problems are quite common. But these problems don’t just exist as a way to exemplify combinatorics. This is also a problem that both Cher from Clueless, and I have in real life. It can be hard to pick out clothes when there are so many options!! So I decided to take it a step further and to actually do a little randomization wherein I define subsets of 8 shirts, 4 shorts, and 4 pairs of shoes. I next randomly generate 3 numbers to pick which clothing items will make up my outfit for the day, and then actually go around in the random outfit.
Here is the clothing sample space:
Since there are 8 shirts, 4 shorts and 4 pairs of shoes there are 8×4×4=128 different outfits possible, assuming that I wear exactly one item from each garment type.
As we discussed in class, not all of these outfits would be something that I would actually want to wear in public. Here are two combinations of shirts and shorts that I probably wouldn’t choose on my own:
I think the shoes pretty much go with everything, assuming that the shirt and shorts go well together. So since there are two shirt and shorts combinations I don’t like, and four pairs of shoes that I could wear with each shirt and shorts combination then there are 2×4=8 outfits I really wouldn’t be excited to wear.
Therefore, the probability that I wouldn’t like an outfit chosen with my random number generator would be 8 ÷ 128 = 0.0625.
Without further ado, here are the numbers that I generated with R:
So I ended up with the seventh shirt, the first pair of shorts and the second pair of shoes:
This is something I would actually feel okay about wearing! I could keep going with this and have a choosing without replacement word problem on my hands, which would be fun too.
As I mentioned above, in the final exam for this class, I made a reference to the fictitious publication Statistician’s Fashions Quarterly. At the time that I wrote that exam question, I wondered what kind of content Statistician’s Fashions Quarterly could even have in it. What would the featured articles be? Who would they interview? What kind of themes could the different issues have?
I think I have now answered those questions for myself.
And of course here I am modeling my randomly generated outfit: