Choosing A Swimming Instructor

[mathdad]

Suppose there are 12 male swimming instructors and 13 female swimming instructors. In how many ways can a student choose a swimming instructor?

Seeking a hint.

Thanks

 

[Dr.Peterson]

How many swimming instructors are there to choose from?

This is probably an example of a problem so simple you don't think they'd bother asking it, and expect something more.

 

[mathdad]

How many swimming instructors are there to choose from?

This is probably an example of a problem so simple you don't think they'd bother asking it, and expect something more.

My friend stated this:

The student has 12 options (the male instructors) and then 13 more options (the female instructors). Since these options don’t overlap, we simply add the number of options together: 12 + 13 = 25 ways to choose an instructor.

 

[khansaheb]

Suppose there are 12 male swimming instructors and 13 female swimming instructors. In how many ways can a student choose a swimming instructor?

Seeking a hint.

Thanks

There are 2 choices: a student can choose

a male instructor

or

a female instructor.

 

[Harry_the_cat]

Forget "maths". It's common sense. There are 25 different options.

 

[mathdad]

Forget "maths". It's common sense. There are 25 different options.

"Common sense" for everyone who understands math.

 

[Dr.Peterson]

"Common sense" for everyone who understands math.

A surprising amount of math is ultimately common sense, and this is an example. The hard part, perhaps, is to understand what the question means, because it isn't quite what you'd expect someone to ask in real life:

Suppose there are 12 male swimming instructors and 13 female swimming instructors. In how many ways can a student choose a swimming instructor?

We could take that in a couple different ways if we had never seen such a question; you might imagine it means "how many different strategies are there for deciding whom to choose"! But in a probability class, the definition of [discrete] probability is "the number of possible (equally likely) ways to accomplish the goal (that is, ways the given 'event' can occur), over the total number of ways something can happen ('outcomes')". So the question is about identifying distinct possibilities (in this case, for the choice of an instructor), in a way that will all be equally likely.

Now, @khansaheb has suggested one wrong way to do this: We could just identify the outcomes as "male" or "female"; then there are only two "ways" to choose. But those are not equally likely, so that is not what they are looking for. Yet it is a conceivable meaning, if you aren't familiar with this type of problem.

All we know is that there are 12 male and 13 female instructors, so the possible outcomes (equally likely if we choose randomly) are those 25 distinct people.

It's quite possible, since you haven't told us, that you haven't yet been told the definition of probability, in which case not everything I've said is available to you. I think authors of introductory books can tend to assume too much of an untaught reader, so confusion is not uncommon.

In the last few days I've run across a similar issue in several students I've worked with literally at all levels -- from arithmetic to linear algebra: In the first chapter of a book, a very basic question is asked, and it is so basic (pretty much common sense) that the student doesn't think it can be that simple, so they instead answer a harder question. This one is similar, in that it really is common sense to count the total number of choices you are given, but you can complicate it by wondering what they mean by "ways".

 

[mathdad]

A surprising amount of math is ultimately common sense, and this is an example. The hard part, perhaps, is to understand what the question means, because it isn't quite what you'd expect someone to ask in real life:


We could take that in a couple different ways if we had never seen such a question; you might imagine it means "how many different strategies are there for deciding whom to choose"! But in a probability class, the definition of [discrete] probability is "the number of possible (equally likely) ways to accomplish the goal (that is, ways the given 'event' can occur), over the total number of ways something can happen ('outcomes')". So the question is about identifying distinct possibilities (in this case, for the choice of an instructor), in a way that will all be equally likely.

Now, @khansaheb has suggested one wrong way to do this: We could just identify the outcomes as "male" or "female"; then there are only two "ways" to choose. But those are not equally likely, so that is not what they are looking for. Yet it is a conceivable meaning, if you aren't familiar with this type of problem.

All we know is that there are 12 male and 13 female instructors, so the possible outcomes (equally likely if we choose randomly) are those 25 distinct people.

It's quite possible, since you haven't told us, that you haven't yet been told the definition of probability, in which case not everything I've said is available to you. I think authors of introductory books can tend to assume too much of an untaught reader, so confusion is not uncommon.

In the last few days I've run across a similar issue in several students I've worked with literally at all levels -- from arithmetic to linear algebra: In the first chapter of a book, a very basic question is asked, and it is so basic (pretty much common sense) that the student doesn't think it can be that simple, so they instead answer a harder question. This one is similar, in that it really is common sense to count the total number of choices you are given, but you can complicate it by wondering what they mean by "ways".

Authors of math, physics, and science textbooks often assume that every person reading a textbook has a solid background in the material. This is a huge mistake leading to confusion and largely discourages students from ever developing an interest in math or physics or whatever.

To me, common sense is not so common. Here is my idea of common sense. Person A and B go to the zoo. Person A tells Person B not to feed the animals or enter their enclosure. Person B disregards what Person A said and pays with his or her life.

Math doesn't work this way. There are many WAYS to solve a problem. Some students understand one WAY better than another.

To be a good teacher, patience and total understanding that not everyone learns at the same pace is crucial. You would be surprised to know that back in my student days, I did very well in math courses. True, I was not an "A" student but managed to pass ALL MY REQUIRED MATH COURSES.

In fact, in the Spring of 1992, I passed MA172 aka precalculus with an "A" minus. It has been decades since I sat in a formal math class. I feel, however, that members on this website (and other math places) don't understand what it means to be away from formal math training for more than 2 decades.