PA-1 / Lesson 1
You might think "how can I solve any problems when we have covered no theory?!". But you can! There are ideas in the classwork exercises that should give you enough intuition already. So use this intuition (or well, any knowledge if you have it...) to solve the problems below. Your solutions may not be perfectly formal at this stage, but it is okay.
Problem 1.
There is a six-sided die with numbers 1, 2, . . . , 6 on its faces. Find the probability of rolling 2 or 5.
Problem 2.
Peter randomly picks a number among 1, 2, . . . , 10. All numbers are equally likely to be picked. What is the probability that he will pick an odd number?
Problem 3.
You toss a fair coin twice (coin has two sides, called “Heads” and “Tails”), what is the probability that the two things you will see will be different?
Problem 4.
Isabella picks a positive integer between 1 and 1000 at random. Do you think she is more likely to get a prime number or not a prime number? Justify your answer.
(Recall that a prime number is a positive integer that has exactly two different positive divisors: 1 and itself. E.g, 2, 3, 13, 101, and 997 are prime numbers, while e.g, 1, 4, 10, 21, and 143 are not)
Problem 5.
Recall the problem from classwork about the Kwacha virus. Imagine that Alla does the test again and, unfortunately, gets a positive result a second time. What is the actual chance that Alla has the virus now? Once again, feel free to use intuitive arguments and ideas similar to the ones presented in the class.
Remember the very first sentence from our lesson: "Probability is a branch of mathematics that can make you really rich nowadays." Since we aim to explain and back up our claims in this educational project, let me share my personal experience.
So, at the moment of writing this text for you, I am actually working as a Quantitative Researcher ("quant" in short). Well, to be more precise, I am doing an internship which is a training period before the actual job. Being a quant involves creating mathematical models, often involving coding, to predict how the price of something will change in the future, with the goal of making money. But no one knows the future; therefore, what we often do is make claims like “with $x\%$ chance the price will behave a certain way in the next $n$ seconds/minutes/days” after analysing the history. This process of analyzing history to make an informed decision about the future is not easy, but it does pay off. In my case the salary is many-many US dollars per week, which could let me buy about one IPhone 12 per day or around one best largest mac book pro per week. And this is just for an internship, i.e., once again, kind of training before the actual job! Note that I am not exceptional — many of my friends have a similar salary doing an internship as a Quantitative Researcher or Trader in other firms. So I am sharing this information not to show off, but to inform you that there are opportunities like this, and there are students taking them.
Being a Quantitative Researcher is one of several jobs that require a strong understanding of probability theory and offer high pay. There are other options, too. While they might not pay as much, they often involve working on fascinating projects. The obvious ones come from AI, which is what you will most probably be doing in the future anyway... :) The reason I have talked about quants (instead of, e.g., ML or cryptography, which I could as well) is to share a specific live example to prove that it's possible to earn a lot of money while being young and studying mathematics!