Homework 6: Bayes' Theorem

1. A desk has three drawers. The first contains two gold coins, the second contains two silver coins, and the third contains one gold and one silver coin. A coin is drawn from a drawer selected at random. Suppose the selected coin is silver. What is the probability that the other coin in the drawer is gold?
2. English and American spellings are rigour and rigor, respectively. A man staying at a Parisian hotel writes this word, and a letter taken at random from his spelling is found to be a vowel. If 40% of the English-speaking men at the hotel are English and 60% are American, what is the probability that the writer is an Englishman?
3. 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. If one knows only that the second card drawn was the two of clubs, find the probability that the first card was diamonds.
4. Next semester, Wisconsin Tech will offer 6 sections of a course called Advanced Probability. Professor X will teach 1 of the 6 sections; Professor Y will teach 2; and Professor Z will teach the remaining 3. Alice needs to take the course in order to graduate in May. She will choose a section randomly by rolling a 6-sided die. She estimates that her probability of passing the course is 80% with Professor X, 60% with Professor Y, and 40% with Professor Z. Suppose that after the semester ends, we learn that Alice did, indeed, pass the course. What is the probability that she was enrolled in Professor X's section?
5. At a card table in a casino in Reno, Nevada, a dealer deals from a standard deck of cards on weekdays and from a double deck (i.e., two standard decks combined) on weekends. A man named Ace chose a day at random last summer to visit this casino. He sat down at the card table and was dealt four cards off the top of a shuffled stack of cards. All four were aces. What is the probability that it was a Tuesday?