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## Discrete Mathematics Test Question

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Part 1: Suppose that F and X are events from a common sample space with P(F) not = 0 and P(X) not = 0.
(a) Prove that P(X) = P(X|F)P(F) + P(X|F¯)P(F¯). Hint: Explain why P(X|F)P(F) =
P(X ? F) is another way of writing the definition of conditional probability, and then use
that with the logic from the proof of Theorem 4.1.1.
(b) Explain why P(F|X) = P(X|F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes
Theorem.
Part 2: A website reports that 70% of its users are from outside a certain country. Out of their users
from outside the country, 60% of them log on every day. Out of their users from inside the country,
80% of them log on every day.
(a) What percent of all users log on every day? Hint: Use the equation from Part 1 (a).
(b) Using Bayes Theorem, out of users who log on every day, what is the probability that they are
from inside the country?

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