PA-4 / Lesson 1
This page lists extra materials and problems for anyone who's finished everything else in this lesson. They're tougher, but still doable—especially if you found the main materials easy.
Problem 1.
Two chess kings: black and white, are randomly placed on two different cells of a chessboard. Clearly describe the probability model that "makes sense" for this problem, and answer the following question: what is the probability that one king attacks another one?
Problem 2.
In a school football tournament there are 8 participating teams, equally good at football. Every game ends with one of the teams winning. Suppose, the tournament has been played, and you don’t know the results.
What is the probability that two random teams, say A and B
Problem 3.
Following up on the previous problem, what is the probability of teams A and B meeting at all in the tournament?
Problem 4.
A line of 100 hundred passengers are waiting to board on a plane. For convenience, let's say that $i$-th passenger has a ticket for place number $i$. Being drunk, the first person takes a seat at random. But all other 99 passengers are sober, so they will go to their proper seat unless it is already take. In that case, they will randomly choose a free seat. What is the probability that the person number 100 will get to his/her own seat?
Problem 5.
Paula shuffles a deck of cards thoroughly, then plays cards face up one at a time, from the top of the deck. At any time Victor can interrupt Paula and bet £1 that the next card will be red (if he never interrupts, he's automatically betting on the last card). What's Victor's best strategy? How much better than 50%-50% of winning can he do? Assume there are 26 red and 26 black cards in the deck.