ECE2191 - Probability and AI for engineers
Probability Theory
Second semester 2024
1. Alice rolls a fair die twice and obtain two numbers X1 = result of the first roll, and X2 = result of the second roll. Find the probability of the following events:
(a) A defined as X1 < X2
(b) B defined as "Alice observes a 6 at least once"
2. Alice and Bob each choose at random a number in the interval [0, 2]. We assume a uniform probability under which the probability of an event is proportional to its area. Consider the following events:
A: The magnitude of the difference of the two numbers is greater than 1/3.
B: At least one of the numbers is greater than 1/3.
C: The two numbers are equal.
D: Alice’s number is greater than 1/3.
Find the probabilities P (B), P (C), and P (A ∩ D).
3. We roll two fair 6-sided dice. Each one of the 36 possible outcomes is assumed to be equally likely.
(a) Find the probability that doubles are rolled.
(b) Given that the roll results in a sum of 4 or less, find the conditional probability that doubles are rolled.
(c) Find the probability that at least one die roll is a 6.
(d) Given that the two dice land on different numbers, find the conditional probability that at least one die roll is a 6.
4. A batch of one hundred items is inspected by testing four randomly selected items. If one of the four is defective, the batch is rejected. What is the probability that the batch is accepted if it contains five defectives?
5. An electrical system consists of identical components that are operational with probability p, independently of other components. The components are connected in three subsystems, as shown in the following figure. The system is operational if there is a path that starts at point A, ends at point B, and consists of operational components. This is the same as requiring that all three subsystems are operational. What are the probabilities that the three subsystems, as well as the entire system, are operational?
6. A system consists of a controller and three peripheral units. The system is said to be “up” if the controller and at least two of the peripherals are functioning.
(a) Find the probability that the system is up, assuming that all components fail inde- pendently with probability 0.1, and the controller fails independently with probability 0.2.
(b) Suppose a second identical controller with failing probability of 0.2 is added to the system, and that the system is “up” if at least one of the controllers is functioning and if two or more of the peripherals are functioning. Find the probability that the system is “up”.
7. You enter a chess tournament where your probability of winning a game is 0.3 against half the players (call them type 1), 0.4 against a quarter of the players (call them type 2), and 0.5 against the remaining quarter of the players (call them type 3). You play a game against a randomly chosen opponent. What is the probability of winning?
8. Consider an experiment involving two successive rolls of a 4-sided die in which all 16 possible outcomes are equally likely and have probability 1/16.
(a) Are the events Ai = {1st roll results in i} and Bj = {2nd roll results in j} independent?
(b) Are the events A = {1st roll is a 1} and B = {sum of the two rolls is a 5}, independent?
(c) Are the events A = {maximum of the two rolls is 2} and B = {minimum of the two rolls is 2}, independent?
9. Die A has 4 red and 2 white faces, whereas die B has 2 red and 4 white faces. A fair coin is flipped once. If it lands on heads, the game continues with die A; if it lands on tails, then die B is to be used.
(a) Show that the probability of red at any throw is 1/2.
(b) If the first two throws result in red, what is the probability of red at the third throw?
(c) If red turns up at the first two throws, what is the probability that it is die A that is being used?
10. A total of 50 percent of the voters in a certain city classify themselves as Independents, whereas 30 percent classify themselves as Liberals and 20 percent say that they are Con- servatives. In a recent local election, 35 percent of the Independents, 60 percent of the Liberals, and 50 percent of the Conservatives voted.
(a) What fraction of voters participated in the local election? A voter is chosen at random.
Given that this person voted in the local election, what is the probability that he or she is:
(b) an Independent?
(c) a Liberal?
(d) a Conservative?
11. [Optional] Many communication systems can be modeled in the following way. First, the user inputs a 0 or a 1 into the system, and a corresponding signal is transmitted. Second, the receiver makes a decision about what was the input to the system, based on the signal it received. Suppose that the user sends 0s with probability 1 —p and 1s with probability p, and suppose that the receiver makes random decision errors with probability ε . For i = 0, 1, let Ai be the event “input was i,” and let Bi be the event “receiver decision was i.”
(a) Find the probabilities P (Ai ∩ Bi) for i = 0, 1 and j = 0, 1.
(b) Find which input is more probable given that the receiver has output a 1. Assume that the input is equally likely to be 0 or 1 and ε = 0.2.
12. [Optional] Let C1, C2, ..., CM be a partition of the sample space S , and A and B be two events. Suppose we know that
• A and B are conditionally independent given Ci, for all i ∈ {1, 2, ..., M }
• B is independent of all Ci’s
Prove that A and B are independent.
13. [Optional] You get a stick and break it randomly into three pieces. What is the probability that you can make a triangle using the three pieces? You can assume the break points are chosen completely at random, i.e. if the length of the original stick is 1 unit, and x, y, z are the lengths of the three pieces, then (x, y, z) are uniformly chosen from the set
{(x, y, z) ∈ R3 : x + y + z = 1 for x, y, z ≥ 0}
14. Find the number of possible 10 character passwords under the following restrictions: (Note there are 26 letters in the alphabet.)
A: All characters must be lower case letters.
B: All characters must be lower case letters and distinct.
C: Letters and digits must alternate and distinct (as in 1w2x9c4u5s or a1b2c3d4e5).
D: All characters must be lower case, distinct and in alphabetical order. (e.g. abfghikmno is allowed, but not bafghikmno).
E: The word can only contain the upper case letters A and B.
F: The word can only contain the upper case letters A and B, and must contain both of them.
G: The word can only contain the upper case letters A and B, and must contain an equal number of each.
15. Find the number of different words that can be formed by rearranging the letters in the following words: (Include the given word in the count.)
A: NORMAL
B: HHTTTT
C: ILLINI
D: MISSISSIPPI
16. Consider 2 classes, the first with N students, and the second with M students. From this group of N + M students, a pair of students (that is, an unordered set of 2 students) is to be selected.
A: How many choices are possible?
B: How many choices are possible if both students come from the first class?
C: How many choices are possible if both students come from the second class?
D: How many choices are possible if the pair includes one student from each class?
17. Luke rolls a die 18 times. What is the probability that each number appears exactly 3 times?
18. The lottery in the city of Problandia works like this: 100 balls numbered 0 - 99 are placed in an urn, and 5 balls are withdrawn, giving an ordered sequence of 5 numbers. Once balls are drawn they are not replaced before the next are drawn. Citizens buy tickets with 5 numbers of their choosing. The jackpot is awarded for matching all 5 numbers in the right order.
A: How many possible outcomes of the lottery drawing are there?
B: Mrs. Bernoulli plays 5 numbers corresponding to the days-of-the-month on which each of her 5 children were born (her children were born on different days). She buys several tickets, one for each of permutations of these 5 numbers. How many tickets does she buy? What is her chance of winning the jackpot?
C: A separate prize is awarded for getting the correct set of 5 numbers, but not necessarily in the right order. What is the probability that any given ticket will win this prize? What is Mrs. Bernoullis probability of winning this prize?
19. How many ways are there to seat 10 people, consisting of 5 couples, in a row of seats (10 seats wide) if
A: the seats are assigned at random?
B: all couples are to get adjacent seats?
20. An urn (box) contains 30 balls, of which 10 are red and the other 20 blue. Suppose you take out 8 balls from this urn, without replacement. What is the probability that among the 8 balls in this sample exactly 3 are red and 5 are blue?
21. Assume a committee of 10 has to be selected from a group of 100 people, of which 40 are men and 60 are women.
A: How many ways are there to choose such a committee?
B: How many ways are there to choose the committee so that exactly half of the members are men?
C: What is the probability that a randomly selected committee of 10 consists of exactly 5 men and 5 women?
22. 90 students, including Michael and Edmund, are to be split into three classes of equal size, and this is to be done at random. What is the probability that Michael and Edmund end up in the same class?
23. Six people get into an elevator at the ground floor of a hotel which has 10 upper floors. Assuming each person gets off at a randomly chosen floor, what is the probability that no two people get off at the same floor?
24. [Optional] An investor has 20 thousand dollars to invest among 4 possible investments. Each investment must be in units of a thousand dollars. If the total 20 thousand is to be invested, how many different investment strategies are possible? What if not all of the money needs to be invested?
25. [Optional] A player is randomly dealt 13 cards from a standard 52-card deck.
A: What is the probability the 13th card dealt is a king?
B: What is the probability the 13th card dealt is the first king dealt?
26. [Optional] (De Méré,s puzzle) A six-sided die is rolled three times independently. Which is more likely: a sum of 11 or a sum of 12? (This question was posed by the French nobleman de Méré to his friend Pascal in the 17th century.)