The first principle is that you must not fool yourself—and you are the easiest person to fool.
—Richard Feynman
What is the probability that two dice rolls sum to 6 given that the first one is 3?
A certain disease impacts 1% of the population. A test is devised which is very sensitive: If you have the disease, it reports positive 99% of the time. However, it has a small false positive rate of 0.5%.
What is the probability that you have the disease given that you get back a positive test? (You should use a calculator.)
Alice is trying to send Bob a one-bit message. She picks either 0 or 1, with both values equally likely, and sends it. If she sent a 0, there is a 5% chance of an error occurring, and if she sent a 1 there is a 10% chance of an error occurring.
Given that Bob received a 1, what is the probability that Alice actually sent a 1?