Minimizing memorization

In third year, I was forced to take Organizational Behavior. There were more than 150 definitions that we were supposed to memorize for the exam. You won't find many people who enjoy memorizing definitions on a Friday night less than me.

On the exam we were going to have a choice of questions: e.g. you would have to choose, say, 10 of 20 definitions. The problem then, was this: how do I memorize the least amount of useless knowledge and still do well on the exam?

I thought about the problem for a while and decided that it was like a game of Keno in a casino.

In Keno there are 80 balls which are numbered 1 through 80. The player chooses 20 numbers and hopes that many of them or none of them will match the 20 numbers randomly chosen by the casino. The odds of choosing all 20 correctly are ridiculously small and the chances of choosing 0 are also very small. It makes sense that some of the numbers you choose will show up, and some won't.

For example, if I choose 20 numbers, then the casino chooses 20 numbers randomly from the 80 available. The chance that I will have chosen 5 numbers that are the same as what the casino chose is 23.3%. By contrast, the chance that I chose 0 correctly is 0.119%. The chance of choosing all 20 correctly is 1 in 3,535,316,142,212,173,800 (or basically zero).

Anyway, the point is that we can use this Keno game to reduce the amount of memorization that we have to do for a test.

Let's say that there are 20 questions, like on my Organizational Behavior exam. I only have to choose 10 questions. The questions are basically "what does this word mean," so you just have to memorize the answer.

We can think of this like a game of Keno. There are some number of total definitions that could be on the exam. For example, in my OB course, there were about 150 definitions. This is like the total amount of numbers in Keno.

Then we need to know how many answers we want to get right. Let's say that I want to do fairly well and I want to get 8 out of 10 right.

In a game of Keno, with 80 numbers and a player choosing 20 numbers, the chance that k of the n numbers chosen by the player match the 20 chosen by the casino is:

Odds in Keno

To generalize this formula to suit our problem, we just change the 80 to be the number of definitions that could be on the exam (i.e. all the stuff you learned in the course). Then n is the number of definitions that I memorized and k is how many I want to get right. We replace the 20 with the number of questions that actually are on the exam.

If we want to get at least a certain mark, like at least 50%, then we add up all of the probabilities above and including that k value.

Then the formula becomes:

Let's use this formula for my upcoming mandatory history course exam. I can say that there are 15 main topics in the course that might have "short answer" questions. On the exam, there will be 10 "short answer" questions and I will have to choose 4 of them. I want to get all 4 right, of course. How much stuff can I avoid learning and still have a 90% chance of getting 4 out of 4?

I try subbing in different values for answersKnown and I find that if I know a lot about seven out of fifteen topics, I will have a 90.0% chance of being able to answer at least four out of the ten questions that will end up on the exam. Since I only need to choose 4, I don't care if I can't answer all ten or even five of ten.

It's interesting to see how much extra work increases the chance that I'll get all four questions. If I know 8 things, I have a 98.1% chance of getting four out of four. Clearly, there is no point in learning any more than just over half of the material in the course for this section of the exam.

It's interesting to note that the chance that I'll be able to answer four questions drops off very quickly. If I know only 5 or 6 things, the chances are 43.4% and 71.3%, respectively.

Criticisms
If you can find a problem with my formula or any flaws in my reasoning, please leave a comment so that I can fix it!

I used the formula to minimize the amount I had to learn in third year Organizational Behavior and ended up with a 94% in the course. (which is also the minimum mark to qualify for an A+) But this doesn't prove that it works! Maybe I just got lucky.

Remember, study smart, not hard!