PA-4 / Lesson 1
Please do not forget to specify your probability model in each of your solutions! It takes 1-2 sentences, but it is a good exercise to not forget them. It is important.
Problem 1.
You pick a number 2, 3, ... or 12 and then roll two fair dice and calculate the total number of dots on them. If the total is equal to the number you pick, you win a dollar. Which number would you choose to maximise your chance of winning and why?
P.S For those know a board game called Catan: this problem is the reason why number $n$, where $n$ is the answer to this problem, is special in that game :)
Problem 2.
If you pick a 9-digit number at random, what is the probability that its sum of digits will be even? (numbers do not start with zero)
Problem 3.
There are two fair dice with some numbers written on them. You roll them and calculate the sum as usual. Is it possible, that all of the outcomes 1, 2, ..., 36 have the same non-zero probability?
Problem 4.
Continuing on the set up with two dice: suppose that each of them has numbers 1, 2, .., 6 as usual, but each of the dice need not be fair (moreover, they need not be identical either). Is it possible that when rolling them, the probabilities of the events "the sum is 2, the sum is 3", ..., "the sum is 12" are all the same and non-zero?
Problem 5.
There are a 100 people and there are 100 personalised letters that need to be sent to them. Suppose you send letters to them at random so that each person gets a letter (but maybe not the one that was meant for him/her). Let $p_k$ denote the probability that exactly $k$ people receive the letters that were meant for them (e.g $p_{100} = \frac{1}{100!}$, where $n!$ stands for the product $n \cdot (n-1) \cdot (n-2)...\cdot 1$). Find the value of \[p_1 + 2p_2 + 3p_3 + ... + 100p_{100} .\]
A gambler's dispute in 1654 led to the creation of a mathematical theory of probability by two famous French mathematicians, Blaise Pascal and Pierre de Fermat. Antoine Gombaud, Chevalier de Méré, a French nobleman with an interest in gaming and gambling questions, called Pascal's attention to an apparent contradiction concerning a popular dice game. The game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money on the occurrence of at least one "double six" during the 24 throws. A seemingly well-established gambling rule led de Méré to believe that betting on a double six in 24 throws would be profitable, but his own calculations indicated just the opposite. This problem and others posed by de Méré led to an exchange of letters between Pascal and Fermat in which the fundamental principles of probability theory were formulated for the first time. Although a few special problems on games of chance had been solved by some Italian mathematicians in the 15th and 16th centuries, no general theory was developed before this famous correspondence.
The Dutch scientist Christian Huygens, a teacher of Leibniz, learned of this correspondence and shortly thereafter (in 1657) published the first book on probability; entitled De Ratiociniis in Ludo Aleae, it was a treatise on problems associated with gambling. Because of the inherent appeal of games of chance, probability theory soon became popular, and the subject developed rapidly during the 18th century. The major contributors during this period were Jakob Bernoulli (1654-1705) and Abraham de Moivre (1667-1754).
And only starting around 1812, thanks to Pierre de Laplace, the probability theory started being used for many other things beyond games of chance, like say the theory of errors, actuarial mathematics, and statistical mechanics.