Venn Diagrams and Formulas

At the math office today, two math teachers were discussing probability. Two things were surprising about it. One, it was a Saturday afternoon part of our winter break, so no teachers should have been on campus. (We’re a dedicated lot, us math teachers.) Two, the topic they were discussing was so simple, and yet, it reveals the real mind-bending character that probability has on us and our students.

Question 1: What is the probability that you draw a heart or a queen from a deck of cards?
Question 2: What is the probability that you roll a die and get a number less than 4 or an odd number?

Both questions are simple enough. The first one is 16/52 (because there are 16 cards which are hearts or queens in a deck). The second one is 4/6 (because you can roll a 1, 2, 3, or 5).

The problems are seemingly the same. Let’s now look at this problem from the perspective of a venn diagram.

picture-1If we want to know the probability that event A or event B occurs, we clearly can see that we have:

P(A\text{ or }B)=P(A)+P(B)-P(A \text{ and } B)

(We have to subtract that last term, because we added that overlapping section twice when we took P(A)+P(B).)

Let’s apply that to our two questions:

Question 1: We have a probability of: P(\text{Heart or Queen})=P(\text{Heart})+P(\text{Queen})-P(\text{Heart and Queen}). Clearly P(\text{Heart})=13/52 and P(\text{Queen})=4/52. What is P(\text{Heart and Queen})? Intuitively — or using our venn diagram — we know it is 1/52. And the answer works out correctly to 16/52.

Question 2: We have a probability of:  P(\text{less than 4 or odd number})=P(\text{less than 4})+P(\text{odd number})-P(\text{less than 4 and odd number}). Clearly P(\text{less than 4})=3/6 and P(\text{odd number})=3/6. What is P(\text{less than 4 and odd number})? Intuitively — or using our venn diagrams — we know it is 2/6. And the answer works out correctly to 4/6.

However, let’s say we wanted to calculate P(\text{Heart and Queen}) and P(\text{less than 4 and odd number}) (the overlapping regions) mathematically? It turns out that there is something fundamental that makes these two problems different. In question 1, the two events (drawing a heart / drawing a queen) are independent. In question 2, the two events (rolling a number less than 4 / rolling an odd number) are dependent. For the first question, you can say that P(\text{Heart and Queen})=P(\text{Heart})P(\text{Queen}) while in the second problem you cannot do that.

Recall that the definition of independence of two events A and B is if P(A|B)=P(A).

Checking the first question for independence, we see that the probability of drawing a heart given that you already have a queen is 1/4, and that is the same as the probability of drawing a heart (1/4). (Similarly, the probability of drawing a queen given that you already have a heart is 1/13, and that is the same as the probability of drawing a queen (1/13).) So the two events are independent.

Checking the second question for independence, we see that the probability of rolling and odd number given that you have rolled a number less than 4 is 2/3, while the probability of rolling an odd number is 1/2. (Similarly, the probability of rolling a number less than 4 given that you’ve rolled an odd number is 2/3, while the probability of rolling a number less than 4 is 1/2.) So the two events are dependent.

The teacher who brought up this problem was grading exams, and one student had calculated P(\text{Heart and Queen})=P(\text{Heart})P(\text{Queen}). And seeing the two problems were almost identical, calculated P(\text{less than 4 and odd number})=P(\text{less than 4})P(\text{odd number}) — which, as we know, isn’t right for dependent events.

What we were discussing is how we could explain to the student that the two situations are different, even though on the surface the questions seem like they are of the same form. In other words, is there a conceptual — non mathematical — way to explain that the first question involves independent events while the second question involves dependent events? It certainly isn’t intuitive, at least not to me.

Analyzing Parametric Equations

I saw a tweet that sumidiot posted on the parametric equations x(t)=t+c*\sin(2t) and y(t)=\cos(t) and spent a good 30 minutes thinking about the value of c which makes the graph intersect itself exactly once. I was going to post about my solution, but I was beaten to the punch. And thank goodness, because there are graphs and everything on sumidiot’s solution, which you should read here.

I still haven’t fully evaluated twitter, which I will post about later once I come to some solid conclusions. But I’m leaning towards liking it. I’ve got lots of good links, anyway. If you want to be my twitter friend, I’m here!

I feel awesome

I handed back a graded problem set back to my multivariable calculus class last week. One student emailed me to meet to go over it with him. We did that today.

But the best part? One of my comments was that there were not enough words and motivation for each mathematical step. His response — totally unprompted by me — was something to the effect of “Now that I had to do all that writing for the Kepler paper, I now think I know what sorts of things I’m needing to explain, that I wasn’t explaining before.”

If there was any smidgen of doubt left in my mind that our Kepler “unit” wasn’t useful, it has been eradicated. The skills I wanted them to pick up? They can now apply them to the rest of the course. HUZZAH!

College Admissions Time!

It’s college admissions time again — and for those who applied early action / early decision, the results are coming in this weekend and next week. I try my best to ignore all of this in the classroom. I know that some students are going through devastating times, while others are so elated they can’t contain themselves. But I don’t know which is which, and honestly, I don’t want to know.

Why? Because I can’t do anything about it.

College acceptances and rejections are, unfortunately, a trial that all seniors have to endure [1]. Having books, counselors, and teachers around telling them “it’ll all be okay” is fine and dandy, but it doesn’t do them any good. Yes, we know it’s true. We know that these students will get in somewhere. We know that years down the road, they won’t be able to imagine going to a different college than the one they went to, because they’ll have made all these friends and had these amazing experiences. We might even say all that to them, in addition to the “it’ll all be okay.” But our words won’t make a difference to them, and we know that too. We say it because we can’t say anything else, because we’re helpless to help in this trying time. Really we say it for ourselves, our own contrived pretense that we can help in a situation where we have no control.

Their hurt is real, immediate, and prevents a broad outlook.

My own story shapes how I feel about this.

As a senior, I applied early to Harvard. I told everyone that it was a long shot and I didn’t think I would get in, and that I didn’t care if I did or didn’t. I applied early just to get it over with. And part of me wanted to believe all that, but a deeper part of me thought that I couldn’t not get in. This wasn’t because I was vain or conceited or thought I was something special. (Trust me, I was incredibly self-effacing in high school.) But the truth of the matter was I honestly couldn’t think of anything I could have done to make my application stronger [2]. Although I said I wouldn’t get in, I thought I would.

You know where this is going… I didn’t get in. I got the small envelope and was devastated.

And the honest truth was: I wasn’t crushed because I didn’t get into Harvard. It wasn’t some place I had always longed to go, or had a sweatshirt from, or anything like that. Harvard was just one of the five schools I applied to. I was devastated because:

(1) I was judged and was deemed not be good enough
(2) I didn’t know why I wasn’t accepted
(3) I had worked so hard in high school and I felt it was all for nothing
(4) What would my friends think of me, because I believed they all expected me to get in

Those thoughts rattled around in my brain for months. Seriously. I tried to shake them off, but couldn’t. All the nice things that people said to me slid right off, because “it’s all going to be okay” didn’t address any of my concerns. It wasn’t until months later when I saw the inner workings of the MIT admissions office — working there as an undergraduate — that I saw the insane number and quality of applications that were coming in. By that time, of course, I was over the college admissions fiasco. I was enjoying my freshman year. But I finally understood the arbitrariness that sneaks its way into college admissions, and comprehended the statement that I had heard from colleges way back when I was a senior in high school: if we threw out all the applications of the students we admitted, and picked another freshman class from the remaining students, we would get a class as strong and successful as the first.

All of this is to say: this time can suck for seniors emotionally. Rejection or deferral from college is complex, because it deals with how we perceive ourselves, how we think others perceive us, and toys with our own notions of self-worth. Our grades aren’t applying. We are. We make it into a story of morality: have we been deemed good enough to enter the hallowed gates?

I wish I could give all my senior who were rejected / deferred / waitlisted early the gift of hindsight and perspective. But I can’t. So I’ll just say “everything will be alright, I promise” and continue on with my job.

[1] Well, actually, I wrote a letter of recommendation for a junior last year to get into a summer program at one particular school, and if she got in this summer program and did well, she was guaranteed acceptance to that school. So this one senior got a free pass.

[2] If you really want to know: near perfect SATs and SAT IIs, straight As in all my classes, 5s on all my APs (which spanned a variety of disciplines), going to the local state university to take multivariable calculus my senior year, doing a number of extra curricular activities which focused on community service, two summers of mathcamp, and participating in a number of different math competitions. (I also had a pretty good social life, believe it or not. I loved high school. And honestly, for the most part, I didn’t join things for college applications… I joined specific clubs because I loved what we did… or because all my friends were involved with them…)

Parent Conferences: Check

I can check parent conferences off my to-do list. I had around 25 this year, and most of them were enjoyable 10-15 minute conversations. About half were parents of students who are doing fabulously in my class. About half were parents of students who need to recognize that their previous method of doing the bare minimum and not seeking help wasn’t going to work anymore. Some parents have a reasonable and healthy outlook on grades. All the parents — this year — recognized that we were both on the same side. One parent even brought me a tin full of candy!

Overall: success.

Students Making Their Own Position/Velocity Scenarios!

I closed up a unit in calculus on position/velocity graphs. Most of my students had horrific memories of physics their freshman year. That teacher, needless to say, is gone. Last year, a number of my calculus students just shut down when we encountered this topic.

This year, I focused a lot on the concepts. One day I showed dy/dan’s graphing stories and that night, I had them each come up with their own problems. For these problems, they needed to draw the velocity versus time graph and the position versus time graph.

Initially I was going to use my favorite on the assessment, however, there were so many hilarious, exemplary problems, I had to type them up and spend the next day using them.

Some of my favorites:

1. Dare-devil Mr. Shah one day decides to go Bungee Jumping. At the top of a mountain, scared, he hesitates 2 seconds, then jumps. He [falls] quickly, eventually reaching terminal velocity at 100 m/s . At the bottom the rope reaches its limit pulling him back up, [coming] to a stop. Mr. Shah smiles.

2. Mr. Shah is riding the elevator to the 4th floor. He waits for the elevator for a bit and then gets on. The elevator goes to the basement to make sure no one is waiting down there [and, of course, no one is there, as always]. It quickly goes back up to the first floor, where 15 seniors try to crowd on. When everyone is in the elevator it heads up to the 4th floor stopping at the 3rd floor to let people off. Finally Mr. Shah reaches the 4th floor and comes over to our calculus class.

3. The Jonas Brothers are walking down the streets of New York City at a strolling rate of 2 mph for 10 minutes as they composed a new song. Suddenly, [student 1] and [student 2] began running at them screaming, at 7 mph. Struggling to find a hiding spot, the brothers run down the block at 8 mph for 5 whole minutes, when they lost the crazy groupies-in-training. Stopping for a break, the boys catch their breath for 5 minutes on a stoop. Walking away when the coast was clear at the same strolling rate as they began with, Nick remarked, “Sorry guys. I’ll try to be less attractive.”

4. A helicopter is taking off. It rises constantly at 200 ft/minute. After rising for five minutes. It stops for one minute to survey the surrounding area. After rising again for 2 minutes, the helicopter is abruptly blown up by a terrorist missile.

5. A man runs from a tiger going at a constant velocity of 3 mph for 1 hour. The tiger gets tired so the man catches his breath for 20 minutes. A rhino appears and begins to chase again and the man picks up speed to 5 mph.

6. You are in an elevator on the top floor (6th floor). Each floor, it picks up more people and it goes slightly faster each time. When it stops on the 2nd floor, so many get in that it breaks and crashes to the basement. People die.

This was a fun class. And almost all my class got pefect scores on the conceptual part of the latest assessment, on this material. They got it1 They really got it!