Homework 5: Law of Total Probability

1. Section 2.1, #2, 4, 13.
2. A multiple choice exam has 5 choices per question. On 75% of the questions, you think you know the answer; on the other 25% of the questions, you just guess at random. Unfortunately, when you think you know the answer, you are right only 80% of the time. Find the probability of getting an arbitrary question right.
3. Scotty has \(n\) keys of which one will open the door to the engine room.
   1. If he tries the keys at random, discarding those that do not work, what is the probability that he will open the door on the \(k\)-th try?
   2. What if he does not discard the previously tried keys?
4. Draw a card. If it's diamonds, place it back in the deck and draw another card. If the first card is not diamonds, then draw a second card without replacing the first card. Find the probability that the second card is the queen or king of diamonds.