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? See more

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.

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