, 2 min read
Probabilities are unnecessary mathematical artifacts
- While mathematically convenient, probabilities can be harmful when solving problems because hardly anyone can think correctly about them. Here is my evidence: The famous Monty Hall problem has confused people for years because it asks us a probabilistic question. To recap the problem is as follows: behind one door out of three is a treasure. You pick a door. An agent picks one of the two remaining doors and tells you that there is no treasure behind it. Thus, two doors are left, the one you picked and the one nobody picked. Which is more likely to hide the treasure? Many people think that the two remaining doors are equally likely to hide the treasure. In fact, the door that you first picked has only a probability of 1/3 to hide the treasure, leaving 2/3 for the other door. Why are people confused? I don’t think it is a difficult puzzle. I believe that it is confusing only because we frame the problem in terms of probabilities. Let us revisit this problem from a determinist point of view. Instead of asking which is more likely to hide the treasure, let us ask a more practical question, free of the probabilistic point of view. With the above scenario, we repeat the experiments for every possible initial setup (all treasure locations). Then we ask which algorithm is best: keep the first door, or offer to switch to the second door. That is, you ask people to solve the problem without any use of probabilities. I believe that far more people would arrive at the right answer in this probability-free setup. It is a purely deterministic challenge. I claim that introducing probabilities is what makes it confusing.
- To make probabilities fit, we sacrifice correctness for mathematical elegance. Most textbooks prove that hash tables have expected constant-time access when hashing is universal. But universality is a probabilistic property which assumes that your hash functions are picked at random from a family of hash functions. Very few computer languages or software libraries implement hashing in this manner. Hashing is overwhelmingly deterministic. Neither your hashing nor your data is random. It is a compelling and elegant analysis, but it is not a correct model for how hash tables work. Thus, textbooks provide the wrong explanation! The correct description of a hash table appears in the Java API documentation: “constant-time performance (…) assuming the hash function disperses the elements properly among the buckets.” See? No probabilities.