PA-1 / Lesson 2
The first four problems below are rather straightforward and nothing tricky. The only crucial thing to keep in mind is to not forget to clearly specify your probability models, where needed. In particular, make sure that the set of simple events is described perfectly! As for the last problem, besides describing the probability space (as always), this is a non-trivial problem with a good idea.
Problem 1.
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice and also participated in anti-nuclear demonstrations. Which is more probable:
1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.
Explain your answer.
Problem 2.
Three professors went to a bar with their papers. When they left in the middle of the night, they were too tired to tell which paper belonged to who, so they took them at random. Find the probability none of them took the right paper.
Problem 3.
If you toss a fair coin four times, what is the probability that there will be an even number of tails? As before, please describe your probability model.
Problem 4.
A two-digit number is picked at random (numbers do not start with zero). Find the probability that this number has no zeros in its decimal representation.
Problem 5.
Mr. Flash tosses a fair coin 100 times. What is the probability that there will be an even number of tails?
(Note that in your probability model you will have a looot of simple events)
Claude Elwood Shannon was an American mathematician, electrical engineer, and cryptographer known as the "father of information theory". The theories he came up with and developed are incredibly useful and important (even nowadays), and in future courses, we will talk about certain parts of his work. Today, let us share one simple yet rather interesting thing that he has done.
So, Shannon believed that people were not a good source of random behaviour and demonstrated this fact with several implementations of machines that he called mind readers. Essentially, it’s an algorithm that allows a computer to play a simple game against a human opponent. In each round of the game, the player has a free choice between two options – left/right or heads/tails, for example. The algorithm attempts to "guess" which the player will choose. And, more often than not, the computer guesses correctly.
How does it work? Not by mind-reading, obviously, but by exploiting the fact that humans do not behave as "randomly" as they think they do. There are many possible algorithms that exploit that fact well, with one of the simplest ones being this: The computer maintains a very simple memory that records the pattern of results of the last two rounds – whether the player won or lost, whether they switched strategy, and then whether they then won or last the following round. The computer then uses this to choose its own best strategy based on the way the player behaved the last time the current pattern occurred. If the computer loses twice in a row using the current strategy, it picks a random response in the next round.
Shannon's "mind reading machines" show that we are not a great source of randomness either. Even a simple algorithm can predict an average human's future choice rather well...