This year I began doing private maths tutoring and I have been learning a lot about the deficiencies in mathematics education that are encountered by our high school students. I am very skeptical about this “new maths” approach. For one thing, the students are not taught to use pen and paper to write out their reasoning and calculation. For another, they make the student rely heavily on intuition. Sometimes intuition helps but other times, intuition can mislead.

Let me illustrate this problem, not original to me.

Assume we have 3 cards with two faces. One card is colored black on both sides, the other is colored white on either side, and the last has black on one side and white on the other. Let us drop the cards in a hat and then choose a card, and then when we get a card,  we choose a side to see at random too. Question: If the side we see is black, what is the probability the other side is black also?  Did you answer 1/2? Your intuition has misled you. You probably thought, by this data we can dismiss the possibility of the card with both white colors (the second card definition)  and just deal with the first and last card. This is not the true situation.

Here is our analysis. The sample space $\Omega = \{ BB, WW,BW \}$ describes the possible color combination of our cards, e.g. BB means one side is black and the other side is black also, etc.  Let $\beta_s =$  “the side we see is black”, $\beta_o =$ “the other side is black”.

So the situation is asking what is $P(\beta_o | \beta_s)$?

$= \frac {P(\beta_o \cap \beta_s)} {P(\beta_s)}= \frac {1/3} {3/6}= 2/3$

$P(\beta_s)$ actually has 3 ways of getting  a black out of 6 ways of getting a face. Then also $P(\beta_o \cap \beta_s)$ is tantamount to getting the first card in our description which is 1 out of 3.

So the moral of the story is that intuition can not be a substitute for formalism. Formalism actually yields a more accurate result. Our intuition is trumped by the formal analysis, which is a better way of approaching the problem.