Problems

There is an elastic band and glass beads: four identical red ones, two identical blue ones and two identical green ones. It is necessary to string all eight beads on the elastic band in order to get a bracelet. How many different bracelets can be made so that beads of the same colour are not next to each other? (Assume that there is no buckle, and the knot on the elastic is invisible).

In one box, there are two pies with mushrooms, in another box there are two with cherries and in the third one, there is one with mushrooms and one with cherries. The pies look and weigh the same, so it’s not known what is in each one. The grandson needs to take one pie to school. The grandmother wants to give him a pie with cherries, but she is confused herself and can only determine the filling by breaking the pie, but the grandson does not want a broken pie, he wants a whole one.

a) Show that the grandmother can act so that the probability of giving the grandson a whole pie with cherries will be equal to \(2/3\).

b) Is there a strategy in which the probability of giving the grandson a whole pie with cherries is higher than \(2/3\)?

In a tournament, 100 wrestlers are taking part, all of whom have different strengths. In any fight between two wrestlers, the one who is stronger always wins. In the first round the wrestlers broke into random pairs and fought each other. For the second round, the wrestlers once again broke into random pairs of rivals (it could be that some pairs will repeat). The prize is given to those who win both matches. Find:

a) the smallest possible number of tournament winners;

b) the mathematical expectation of the number of tournament winners.

Three friends decide, by a coin toss, who goes to get the juice. They have one coin. How do they arrange coin tosses so that all of them have equal chances to not have to go and get the juice?

Klein tosses \(n\) fair coins and Möbius tosses \(n+1\) fair coins. What’s the probability that Möbius gets more heads than Klein? (Note that a fair coin is one that comes up heads half the time, and comes up tails the other half of the time).

Some randomly chosen people are in a room. A mathematician walks in and says that the probability that there exist at least two people with the same birthday is just over \(50\%\). How many people are in the room?

Imagine there’s a disease called ‘mathematitis’ which \(1\%\) of people have. Doctors create a new test to discover whether people have mathematitis. The doctors fine-tune the test until it’s \(99\%\) accurate - that is, if a person \(A\) has it, then \(99\%\) of the time the test will say that \(A\) has it, and \(1\%\) of the time the test will say that \(A\) doesn’t have mathematitis.

Additionally, for person \(B\) who doesn’t have the disease, \(99\%\) of the time the test will correctly identify that \(B\) doesn’t have it - and the other \(1\%\) of the time, the test will say that \(B\) does have mathematitis.

Suppose you don’t know whether you have mathematitis, so you go to the doctors to take this test, and the test says you’ve got it! What’s the probability that you do actually have the disease?

Imagine that people are equally likely to be born in each of the \(12\) months. How many people do you need in a room for the probability that some two are born in the same month to be more than \(50\%\)?

Some doctors make a new test for the disease ‘mathematitis’ which is even better. This new test is \(99.9\%\) accurate - meaning that \(99.9\%\) of the time when someone has the disease, the test will say so. And when someone doesn’t have the disease, \(99.9\%\) of the time the test will say that they don’t have it.

Paul goes to the doctor and test positive for mathematitis. What’s the chance he actually has mathematitis? Recall that \(1\%\) of the population has mathematitis.

Consider the following dice below:

That is, the green die on the left has sides \(2,2,4,4,9,9\), the red die in the middle has sides \(1,1,6,6,8,8\) and the blue die on the right has sides \(3,3,5,5,7,7\). In each of the dice, each side is equally likely to appear.

Imagine we both roll one die, and whoever gets the higher score wins. If I choose the green die, what die should you choose?