Problems

There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.

One of \(n\) prizes is embedded in each chewing gum pack, where each prize has probability \(1/n\) of being found.

How many packets of gum, on average, should I buy to collect the full collection prizes?

Every evening Ross arrives at a random time to the bus stop. Two bus routes stop at this bus stop. One of the routes takes Ross home, and the other takes him to visit his friend Rachel. Ross is waiting for the first bus and depending on which bus arrives, he goes either home or to his friend’s house. After a while, Ross noticed that he is twice as likely to visit Rachel than to be at home. Based on this, Ross concludes that one of the buses runs twice as often as the other. Is he right? Can buses run at the same frequency when the condition of the task is met? (It is assumed that buses do not run randomly, but on a certain schedule).

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?