EE3025 - Statistical Methods in ECE
Spring 2024
Homework 2
Due Friday, Feb. 9, 2024, 5:00 PM
Problem 1
A $100 note is hidden in one of the 10 gift boxes, while the other nine boxes are empty. Alice and Bob keep choosing one box at a time alternatively and open it. Once the box with the note is opened, the money belongs to the person who’s picked the box. Let Alice chooses the first box, and Bob opens the second one (if the first box is empty), and so on. We want to examine which player has a better chance to win, Alice or Bob.
(a) If you play this game, do you prefer to play in Alice’s role (first to pick a box) or Bob’s role (who plays after Alice)? No calculation is needed. Just answer based on your intuition!
(b) What is chance that Alice wins in her first attempt?
(c) What is the probability that Bob wins in his first turn?
(d) Compute the probability that Alice wins in her second choice.
(e) Compute the probability that Alice eventually wins the game.
(f) What is the probability that Bob eventually wins the game.
(g) Answer to part (a) again. Is your initial guess supported by your calculations? Explain.
Problem 2
People in the country are dealing with a new virus with infection rate of 0.1%, i.e., on average 1 out of 1, 000 people are infected. There is a test to check whether a person has the disease. The test is not absolutely accurate. In particular, we know that
• the probability that the test result is positive (suggesting the person has the disease), given that the person does not have the disease, is 5 percent (false positive);
• the probability that the test result is negative (suggesting the person does not have the disease),given that the person has the disease, is 10 percent (false negative).
A randomly selected person is tested positive. What is the probability that the person has the disease?
Problem 3
There are 3 appetizers, 8 main dishes, and 5 desserts on a restaurant menu.
(a) In how many different ways one can choose a full dinner course in this restaurant?
(b) On Friday night the appetizer and main dish menus are mixed and has a total of 11 choices. One can pick one for the appetizer and a different one for the main dish. How many different full courses one can have?
(c) On a special event on Saturday night the chef prepared 4 new dishes and offered them on both the appetizer and main dish menus. How different combinations you can make if you want your main dish to be different from your appetizer?
Problem 4
A car plate in the state of Minnesota consist of three digits followed by three characters.
(a) Find the total number of plates can be issued by the MN DMV.
(b) Find the number of plates who have an A at their 4th position.
(c) Find the number of plates who have at least one A.
(d) Find the number of plates with three consecutive three digits in increasing order, i.e., 012, 123, . . . , 789.
(e) How many plates with no repeated symbols (digits/letter) does exist?
(f) How many plates with letters in {U,V,W,X,Y, Z} does exist?
Now, assume DMV issues the plate according to a uniform. distribution,i.e., issuing all possible plates are equally likely.
(g) What is the chance of getting a plate with at least one A?
(h) What is your chance to get a plate with three letters consistent with your F.M.L. (first name, middle name, lastname) in the right order?
(i) What is the chance of seeing a plate with first digit being (strictly) less than the second one?
(j) A dirty car just passed by yours, and you only saw a 4 at its second digit. What is the probability that its first digit being (strictly) less than the second one?
(k) If someone tells you that in her car’s plate the first digit is strictly less than the second one, what is the chance that her second digit is 4?