Probability, Random Processes and Inference

Transcription

1 INSTITUTO POLITÉCNICO NACIONAL CENTRO DE INVESTIGACION EN COMPUTACION Laboratorio de Ciberseguridad Probability, Random Processes and Inference Dr. Ponciano Jorge Escamilla Ambrosio

2 Problem 1 1. A spam filter is designed by looking at commonly occurring phrases in spam. Suppose that 75% of is spam. In 15% of the spam s, the phrase free money is used, whereas this phrase is only used in 3% of non-spam s. A new has just arrived, which does mention free money. What is the probability that it is spam? 2

3 Solution Problem 1 Let S be the event that an is spam and F be the event that an has the free money phrase. By Bayes Rule: P S F = P F S P S P F With P(S) = 0.75; P(F S) = 0.15; P(F S c ) = 0.03, results in: P F S P S P S F = P F S P S + P F S c P S c (0.15)(0.75) = (0.03)(0.25) = =

4 Problem 2 2. Let A and B be independent events. Let the events A c and B c be the complements of the events A and B. Verify that the events A and B c are independent. Conclude directly from this result that the events A c and B c are also independent. 4

5 Solution Problem 2 5

6 Problem 3 3. A drunkard removes two randomly chosen letters of the message HAPPY HOUR that is attached on a billboard in a pub. His drunk friend puts the two letters back in a random order. What is the probability that HAPPY HOUR appears again? 6

7 Solution Problem 3 7

8 Problem 4 4. A man has 5 coins in his pocket. Two are double-headed, one is double-tailed, and two are normal. The coins cannot be distinguished unless one looks at them. a. The man shuts his eyes, chooses a coin at random, and tosses it. What is the probability that the lower face of the coin is heads? b. He opens his eyes and sees that the upper face of the coin is a head. What is the probability that the lower face is a head? c. He shuts his eyes again, picks up the same coin, and tosses it again. What is the probability that the lower face is a head? d. He opens his eyes and sees that the upper face is a head. What is the probability that the lower face is a head? 8

9 Solution Problem 4 Let D denote the event that he picks a double-headed coin, N denote the event that he picks a normal coin, and Z be the event that he picks the double-tailed coin. Let H Li (and H Ui ) denote the event that the lower face (and the upper face) of the coin on the ith toss is a head. 9

10 Solution Problem 4 10

11 Problem 5 5. An electrical system consists of identical components, each of which is operational with probability p, independent of other components. The components are connected in three subsystems, as shown in the figure below. The system is operational if there is a path that starts at point A, ends at point B, and consists of operational components. What is the probability of this happening? 11

12 Problem 5 12

13 Solution Problem 5 The system may be viewed as a series connection of three subsystems, denoted 1, 2, and 3 in the figure. The probability that the entire system is operational is p 1 p 2 p 3, where p i is the probability that subsystem i is operational. Using the formulas for the probability of success of a series or a parallel system, we have: and 13

14 Problem 6 6. Gambler s Ruin Problem. Two gamblers, Gambler A and Gambler B, play repeatedly. In each round A wins 1 dollar with probability p or loses 1 dollar with probability q = 1 p (thus, equivalently in each round B wins 1 dollar with probability q = 1 p and loses 1 dollar with probability p). We assume initially A has i dollars and B has N i dollars. The game ends when one of the gamblers runs out of money (in which case the other gambler will have N dollars). Find p i, the probability that A wins the game given that he has initially i dollars. 14

15 Solution Problem 6 15

16 Solution Problem 6 16

17 Problem 7 7. A professor gives only two types of exams, easy and hard. You will get a hard exam with probability The probability that the first question on the exam will be marked as difficult is 0.90 if the exam is hard and is 0.15 otherwise. What is the probability that the first question on your exam is marked as difficult? What is the probability that your exam is hard given that the first question on the exam is marked as difficult? 17

18 Solution Problem 7 18

19 Problem 8 8. There are 100 equally spaced points around a circle. At 99 of the points, there are sheep, and at 1 point, there is a wolf. At each time step, the wolf randomly moves either clockwise or counterclockwise by 1 point. If there is a sheep at that point, he eats it. The sheep don t move. What is the probability that the sheep who is initially opposite the wolf is the last one remaining? 19

20 Solution Problem 8 20

21 Problem 9 9. A thief is chased by a police car and reaches a crossing that branches into three possible streets A, B and C, such that the latter two are so narrow that they do not fit a police car. The thief is so nervous that all of his choices are made uniformly at random. If he runs down street A, he is captured since the end of the street is cut by another police patrol. If he runs down street C he escapes since there is no surveillance there. If he runs down street B he finds that the street splits into two lanes: the BA, which leads to street A and the BC which leads to the street C. Calculate: (a) The probability that the thief is captured? (b) What is the probability the thief used lane BC given he escaped? 21

22 Solution Problem 9 22

23 Problem A batch of 100 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? 23

24 Solution Problem 10 Let A be the event that the batch will be accepted. Then A = A 1 A 2 A 3 A 4, where A i, i = 1,, 4, is the event that the ith item is not defective. Using the multiplication rule, we have: 24