Question

Q1)

Consider two events P and Q.

a. Write the general formula used to calculate the probability that either event P occurs or Q occurs or both occur.

b. How does this formula change if:

i. Events P and Q are disjoint (i.e., mutually exclusive of each other).

ii. Events P and Q are nondisjoint events that are statistically independent of each other.

iii. Events P and Q are nondisjoint events that are statistically dependent of each other.

Q2)

Rewrite the following stochastic events in mathematical notation using a suitably defined random variable:

a. The probability that the deflection at the most critical element of a certain structure exceeds 1.80 inches is 0.85.

b. Of every 100 road construction projects, 28 experience two or more fatalities.

c. Twenty percent of the time, a certain aging engineering system component breaks down at least four times during operation every year.

d. Of all college students 75% read their campus newspaper at least once a week.

@3) and Q4) in pic

The following problems are from the course's textbook: Introduction to Civil Engineering Systems, Samuel Labi, Wiley, 2014 (ISBN: 9780470530634)) Question 1: Exercise 3, Textbook page 162. Question 2: Exercise 6 on Text page 163 Question 3. The probability that Joe is late for work is 0.20. The probability that Joe drives fast to work is 0.40. The probability that Joe performs well at work is 0.75. Choose 3 letters, say L, F, W, to denote each of these events. If Joe is late, the probability that he drives fast to work is 0.9. Assume that his punctuality or lateness to work does not influence whether he performs well at work that day. Furthermore, 30 out of every 100 days, he drives fast to work and does a good job in the same day. (a) For all paired combinations of these two events, comment on their mutual exclusivity, statistical dependence, or statistical independence. Use the table below Pair Property (circle one) L and F Mutually exclusive/Statistically dependent/Statistically independent L and W Mutually exclusive/Statistically dependent/Statistically independent Mutually exclusive/Statistically dependent/Statistically independent F and W (b) Using the general probability equation and your prescriptions in (a) above, find the probability that on any randomly selected day,.. (i) either he is late for work or he drives fast to work or both happen. (ii) either he is late for work or he performs well at work or both happen. (iii) either he drives fast to work or he performs well at work or both happen Question 4: A. Indicate which of the following random variables are discrete and which are continuous: i. Whether or not users of a civil engineering system are satisfied with the system: ii. The performance of a hydraulic system in terms of volume of water pumped per hour Rewrite the following stochastic events in mathematical notation using a suitably-defined random variable i. B. Of every 100 road construction projects, 28 experience two or more fatalities. ii, 20% of the time, a certain aging engineering system component breaks down at least 4 times during operation every year C. write in everyday English, the stochastic events that are represented by the probabilities of the following random variables: i. a) p(Y>7)-0.33 Yis a random variable representing the weekly number of hours worked by a randomly selected Catholic University student. ii. b) p(F 2 2) 0.85 Fis a random variable representing the number of times a hydraulic system gets overloaded

Answer 1

1(a)

The general formula of probability when either of two event occur is given as

P(either A or B occurs) = P(AUB) =P(A)+ P(B)

1(b)

(i) when two events are mutually exclusive

P(A ∪ B) = P(A) + P(B).

(ii) when both events are statically independent to each other then

P (P ∩ Q) = P (P) . P (Q)

but for two disjoint event

P (P ∩ Q) = 0

(iii) when both events dependent to each other

when occurrence of one event affect the other, then it is called dependent events

P(A and B) = P(A) · P(B|A)

where P(B|A) probability of A under condition B. ( conditional probability)