Problem 2: Let's make a deal This problem is named after the host of the long running TV show Let's make a deal. There has been a vigorous debate about what the correct answer is! A contestant must choose one of three closed doors. There is a prize (say a car) behind one of the three doors. Behind the other two doors, there is something worthless (traditionally a goat). After the contestant chooses one of the three doors, Monty opens one of the other two, revealing a goat (never the car!!). There are now two closed doors. The contestant is asked whether she would like to switch from the door she initially chose, to the other closed door. The contestant will get whatever is behind the door she has finally chosen. Should she switch?Problem 12: Multiple choice This erercise consists of 4 multiplechoice problems. The first problem requires you to have Ercel installed on your device. If you do not have Excel, please download it here: at tps: // offics365. uchicago. edu/ 12.1 Download the MLB data (RunsPeerme.csv) and open it in Excel. Based on this data, how many runs per game (R/ G) did Chicago (National League) earn over the season? (a) 6.64 (b) 6.266 (c) 4.72 (6;. 6.256 12.2 Suppose X is a random variable. What is P(2 2) (ii) P[X > 2 + v') (iii) P(X = 3} (iv) Express P(8 5 X E 4) in terms of Z, the standard normal random variable. Problem 8: Raw materials A company can purchase raw material from either of two suppliers and is concerned about the amounts of impurity the material contains. A review of the records for each supplier indicates that the percentage impurity levels in consignments of the raw material follow normal distributions with the means and standard deviations given in the table below. The company is particularly anxious that the impurity level in a consignment not exceed 5% and want to purchase from the supplier more likely to meet that specication. Which supplier should be chosen? Mean Standard Deviation Supplier A 4.4 0.4 Supplier B 4.2 0.6 Problem 9: Breakfast Sally gets a cup of coffee and a mufn every day for breakfast from one of the many coffee shops in her neighborhood. She picks a coee shop each morning at random and independently of previous days. rThe average price of a cup of coffee is $1.40 with a standard deviation of $0.30, the average price of a mufn is $2.50 with a standard deviation of $0.15, and the two prices are independent of each other. (a) What is the mean and standard deviation of the amount she spends on breakfast daily? (b) What is the mean and standard deviation of the amount she spends on breakfast weekly [7 days)? Problem 11: Settlers of Catan In the game Settlers of Union, you begin the game by picking a starting spot where three hex pieces meet together. A hex piece is a sixsided piece on the game board with a number attached to it (see below). For example, the blue circle in the picture below touches three he); pieces with the numbers 11, 11, and 3. At the start of each turn, you roll two dice together, and what happens next depends on the sum of these two dice. Without going into the details, let 's simplify things as follows: a If your starting spot is touching a tile whose number equals the sum of the two dice, good things happen; a Otherwise, bad things happen. So, you want to pick a starting spot that will get rolled as often as possible. In a recent game, my friend chose to start as (as a team) at the intersection of (11, 11, 3). We ended up losing this game. Using probability, give a likely explanation as to why we lost the game. You do not need to consider the starting positions of the other players or any other aspects of the game stateI claim this was not a factor in this particular case. Hint. You do not need to compute anything to answer this question, nor do you need any precise probabilities. Just good ole counting