In the last column, I presented two puzzles that could be solved in different ways. The first one is a card problem:

I show you a stack of 18 cards. 6 of them are black and 12 are red. All of the black ones are face cards, and half of the red ones are face cards. I allow you to draw one card from the shuffled deck. You cannot see the face of the card or in any other way bias the outcome. What is the probability that you will draw a card that is either black, or a face card, or both?

For me, the easiest way to solve this one is to recognize that the only cards excluded in the answer are red non-face cards. Since we know there are 6 of them, then the probability of getting a red non-face card is 6/18 or 1/3. Therefore the answer to the question is 2/3.

You can also rather easily derive the same answer by combining the probabilities for actually getting the required condition, but it does take some more computation, and if the puzzle were any more complicated, this would take much more work.

The second problem had the possibility of logical ambiguity depending on the assumptions made.

Assume you have two tables. One is round, and the other is square with distinctly different four sides. You are hosting a party and a total of eight people will attend. You want to seat four of them (A, B, C, D) at one table and four (G, H, I, J) at the other. But you are not sure of how you want them to be organized. How many distinct seating arrangements can you have for each table?

Consider the square table first. This is a simple permutation computation. There are 4 times 3 times 2 = 24 ways. That is pretty straightforward. However, note that we have made the implicit assumption that one person sits per side. That is a reasonable assumption, but not strictly stated. We would get different answers if we assume that 1, 2, 3, or 4 persons could sit on a side.

The circular table is even trickier to nail down. I did not specify the size of the table of the size of the people. So in a certain sense, the most correct answer is that there are an infinite number of ways to seat the second group of 4 guests. However, if we assume a typical sized table and typically sized guests, and further assume that by different arrangements we really mean different ordering of who is sitting next to whom, then we get another permutation problem in which only the order and not the distance is important. That being the case, seat one of the four. Place the remaining three relative to the first one. This means we are looking at the total number of permutations of three guests. This is 3 times 2 – 6 ways.

In both cases, we also implicitly assumed that no one was sitting on another guest’s lap, that might or might not be a reasonable assumption based on your own personal experience.

These two examples demonstrate many things. One of the most important is how difficult it is to truly specify a puzzle accurately. In fact, very few people are really interested in specifying problems exactly. If we all took the time to remove implicit assumptions from our communications, nothing would get done. Far from aiding in accurate transmission of ideas, insisting on assumption-free communication eventually stops communication. Whether it is nerve impulses across synapses, people talking, or countries negotiating, plausible assumptions are critical to effective communication.

Which lead to an interesting question. In normal communication between speakers with the same native language, what is the optimum frequency of implicit assumptions? Too many will definitely lead to mis-communication, and too few will lead to inflated documents (cf legalese) and inattention with resulting loss of data transmission. We know that some communication can take place, and the rate drops at both ends of the assumption spectrum. Therefore, by a slight mis-application of Rolle’s theorem, there must be at least one optimum, and that is not at zero assumptions.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.