Conditional probability is the art of updating probabilities based on given information. What is the probability that the sidewalk is wet? And what if we know that it rained a few hours ago?
You spin a spinner with equal probability of landing on each of the numbers 1 through 10. Your friend covers up the number it lands on, but you can tell that it only has one digit (and therefore can't be 10). What is the probability it landed on a 5?
Suppose that Phil is a compulsive liar. Every statement he says has a 75% probability of being a lie. He flips a fair coin and tells you it's a head. Is the coin flip more likely to have been a head or a tail?
Phil is still a compulsive liar whose every statement has a 75% probability of being a lie. However, we know that he is aware of the winning number in a lottery that consists of choosing a single integer from 1 to a million. He says the winning number is 123. If you were to enter the lottery, which number should you pick to maximize the probability of winning?
Note: if Phil decides to lie about the winning number, he will pick any incorrect but plausible number with equal probability.
Suppose that in a given population of people everyone is either good or bad at jumping, and good or bad at running. These are independent of each other (so knowing if someone is good at jumping tells you nothing about how they are at running). An athletic training camp only accepts people who are either good at running or good at jumping (possibly good at both).
You meet a random person who went to the camp, and you find out they are good at running. Are they more, less, or equally likely to be good at jumping than someone random at the camp whose running ability you don't know?
Note: Assume that there is at least one person good at only running and one person good at only jumping at the camp.
You flip two coins and your friend tells you that both landed on the same side. Conditional on this information, what is the probability that at least one of the coins landed on heads?