In a box, there are 10 white and 15 black balls. Four balls are removed from the box. What is the probability that all of the removed balls will be white?
There are three boxes, in each of which there are balls numbered from 0 to 9. One ball is taken from each box. What is the probability that
a) three ones were taken out;
b) three equal numbers were taken out?
Henry wrote a note on a piece of paper, folded it two times, and wrote “FOR MOM” on the top. Then he unfolded the note, added something to it, randomly folded the note along the old folding lines (not necessarily in the same way as he did it before) and left it on the table with random side up. Find the probability that “FOR MOM” is still on the top.
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).
To test a new program, a computer selects a random real number \(A\) from the interval \([1, 2]\) and makes the program solve the equation \(3x + A = 0\). Find the probability that the root of this equation is less than \(0.4\).
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.