Embracing the Beauty of Math

There are many reasons why I love math. I love that there are often many different correct ways of getting to a solution. I love that there is, at least most of the time, an actual solution and the satisfaction of finding it. I love that math can be abstract, that it can be connected in concrete ways to the world around us, and that it sometimes can do both. And I love combining the analytical with the creative in pursuit of a solution.

Through my job as Director of the CEMC, I get to try to solve many math problems every year. I would like to share one of my favourite recent problems and to touch on some of the reasons for which it stands out to me:

In an experiment, 1000 people receive Medicine A and 1000 different people receive Medicine B. The 2000 people are asked whether they have severe side effects, mild side effects, or no side effects. The following information is obtained from the experiment:

1. The probability that a random person has severe side effects is $\frac{3}{25}$.
2. The probability that a random person with severe side effects was given Medicine A is $\frac{2}{3}$.
3. The probability that a random person who was given Medicine A has severe or mild side effects is $\frac{19}{100}$.
4. The probability that a random person who was given Medicine B has severe or mild side effects is $\frac{3}{20}$.

What is the probability that a random person with mild side effects was given Medicine B?

This problem appeared in Part A of the 2021 Canadian Senior Mathematics Contest. You could look up the solution online and stop reading right now. But, as a mathematician, you are curious and prioritize figuring things out rather than looking up answers.

One reason that this problem is memorable to me is that it appeared at a time when the notion of testing medicines of various kinds was on our minds, in ways that it had never been before. A second reason that this problem has stuck with me is that it was easier to solve using simpler mathematics than it was using sophisticated mathematics. This is a good reminder of the fact that interesting and hard to solve problems do not necessarily involve more advanced concepts.

Since I don't want to spoil your fun, I want to “translate” the problem and then set up the start of a solution that you can then finish.

What do I mean by translate? We can often lose sight of the fact that mathematics is a language and individual words carry lots of meaning at many levels. Is there a specific word that jumps out at you in the text of this problem because of its levels of meaning? For me, it is the word “probability”. The study of probability starts with simple concepts and can become incredibly advanced. When we marked this problem in December 2021, students who tried to use more advanced results from probability generally struggled to make it through their solutions, while students who translated the word “probability” into a simpler concept had a much easier time.

So what does “probability” mean here? Fundamentally, probability is a ratio or a proportion, and in this problem we're not looking at the probability of future events, but looking back at data that we already have. Let's re-write given information in this problem by replacing phrases like “probability that a random person” with “proportion of all people”. Try reading this new version:

In an experiment, 1000 people receive Medicine A and 1000 different people receive Medicine B. The 2000 people are asked whether they have severe side effects, mild side effects, or no side effects. The following information is obtained from the experiment:

1. The proportion of all people in the experiment with severe side effects is $\frac{3}{25}$.
2. The proportion of all people with severe side effects who were given Medicine A is $\frac{2}{3}$.
3. The proportion of all people given Medicine A with severe or mild side effects is $\frac{19}{100}$.
4. The proportion of all people given Medicine B with severe or mild side effects is $\frac{3}{20}$.

What is the proportion of all people with mild side effects who were given Medicine B?
 

This version feels significantly easier than the previous one to solve, even though it is exactly the same mathematically. This highlights a powerful problem solving technique: convert a problem into an equivalent one that is easier to understand and to solve.

So where do we go from here? Let's use the information that we are given in a sequential way, and keep track of it in a visual way. (Two more important problem solving techniques!) Since 2000 people are asked about side effects and $\frac{3}{25}$ of these people have severe side effects, then the number of people with severe side effects is

$\frac{3}{25} \cdot {2000} ={240}$ (This is the information in (1).)

Since 240 people have severe side effects and 23 of them were given Medicine A, then the number of people who were given Medicine A that have severe side effects is $\frac{2}{3} \cdot {240} ={160}$ (This is the information in (2).) 

This means that the number of people who were given Medicine B and have severe side effects is $240 - 160=80$

We can put this information into a chart:  

$240-160=80$

Take some time now to complete the 6 missing entries in the table using the rest of the given information before worrying about the actual question that we are asked. Once you've done this, look at the question that we are asked and see if you can determine which numbers in the table you need to compare to answer the given question.

Now that you've thought about the problem and worked on a solution, this would be a good time to find and read the official solution. Can you find a different correct solution?

We could leave things here, having done the mathematics and reached a satisfying endpoint. But because there is a real world aspect to the problem, it's worth asking some other questions. If the data here were real, what would this tell us about Medicine A and Medicine B? While we don't have data on which Medicine is better at its intended function, we could talk about which is better from the standpoint of side effects. But what does “better” mean? Are we qualified to interpret this data in this way?

This is another amazing property of mathematics as a discipline — it interacts with and intersects many non-mathematical disciplines. People who “do mathematics” in the real world invariably find themselves working with doctors or chemists or psychologists (to name just three other professions) to answer questions with real and immediate impact on people. Remember this the next time you sit down to solve a problem — the mathematics that you are learning and doing is training your brain to be able to make a tangible difference to the world around us.

Article by Ian VanderBurgh