17 September 2011

An example of bad mathematics textbooks for schoolchildren

A friend of mine sent me the following entry from a mathematics textbook for schoolchildren in the US around the age of 12. Here is a question asked, together with the suggested solution:
Question: Tell whether the following events are dependent or independent. If they are independent, find the probability that both events occur.
Event C: Choosing the letter F from a bag containing the alphabet.
Event D: Choosing the letter V from a bag containing the alphabet after already choosing F and not replacing the letter.
Solution: Events C and D are dependent events. Once a letter has been picked from the bag and not replaced, it changes the probability of picking another letter from the bag.
Here is the problem with the way the question is formulated. In defining event D, the writer of this has inserted the description of the sample space. Instead of clearly defining the experiment first and then the events, he/she mixed the two things together in defining the event D. The result is confusion. The correct way of stating the problem is:
We have a bag containing the 26 letters of the alphabet. We pick a letter at random, put it away, and then pick another letter at random and put it away. Define the following events:
Event C: The first letter we pick is F.
Event D: The second letter we pick is V.
Question: Determine whether the two events are independent or not.
The suggested solution is even more confusing. It says that events C and D are dependent (correct, provided you have understood what event D is, i.e., what the author wanted to say but did not say), but the explanation given is not very good: "Once a letter has been picked from the bag and not replaced, it changes the probability of picking another letter from the bag." It changes the probability of what? What does the changing of a probability have to do with the definition of dependence?

Let's see first what the correct solution is:
P(C) clearly equals 1/26.
P(D) also equals 1/26.
The reason is: From the definition of event D, as an event which has nothing to do with whatever we picked at the first pick, and because of symmetry, we can see that P(D)=1/26 as well. What I mean by this is the following: if, say, the letters were arranged in a random order on a line and defined event C to be the leftmost letter, while D the right most one, then it would have been even clearer that P(C)=P(D). This is why the two probabilities are the same.
As for the probability P(C & D) of their intersection we have 
P(C & D) = P(C) P(D | C) = (1/25) x (1/26).

The reason is this:  P(D | C) can be computed on the new sample space containing 25 letters because we are conditioning on having picked the letter F first. Since there are 25 letters remaining, we have P(D | C) = 1/25.
Since P(D & C)  ≠  P(D) P(C), it follows that C and D are dependent events.
A clearly stated problem, and a logical solution is the only way that the kids in the school can learn something which they can subsequently find useful. The kids will remember, even subconsciously, these kind of things and if they have been taught incorrectly, they will have to unlearn everything when they go to the university (provided--this is an assumption--that their university teachers know how to teach or bother to do so).

No wonder why when people ask me what I do for living and I say mathematics they invariable give the same response: "Oh, I was so bad in mathematics at school." "I understand", I reply. "but, most likely, so was your teacher".

P.S. My friend also told me the following idiotic question the kids get in their mathematics class:
Question: What do you call the shape whose area is given by the formula L⋅H?
There is no end to idiocy in this world. 

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It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant