Conditional Probablity in stat
Consider your best friend. Suppose that on any given day, your best friend wakes up happy with probability .8 and unhappy with probability .2. Let X = 1 if your best friend is happy today and X = 0 otherwise. In practice, you do not know whether your best friend is happy today, but there are indicator. One indicator is smiling. If your best friend is happy, (s)he smiles with probability 1 and otherwise your best friend smiles with probability .5. Let Y = 1 if your best friend smiles today and Y = 0 otherwise.

Suppose that your best friend smiles today. What is the probability that your best friend is happy today given that your best friend smiles today, i.e., what is the probability of X = 1 given Y = 1?

Suppose that your best friend is out of town and that you therefore do not see your best friend. What is the probability that your best friend is happy today, i.e., what is the probability of X = 1?
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