Question

1. The Australian Maritime Safety Authority has found that 10% of ships have navigation faults (N) and 20% have other safety faults (F), such as non–compliance with regulations for fire extinguishers. Six percent of ships have both faults. A ship is selected at random.
(a) Show the sample space and the events N and F as a Venn diagram.

(b) What is the probability the ship has either N or F?

(c) What is the probability the ship has neither N nor F?

(d) What is the probability the ship has N given that it has F? (e) What is the probability the ship has F given that it has N?

2. An email filter sends some incoming messages to Trash. However, it is not reliable. If a message is not trash it gets sent to Trash with a probability of 0.04. If a message is trash it is not sent to Trash with a probability of 0.10. Suppose 30% of incoming messages are trash.
(a) Draw a tree diagram to show the sample space.

(b) What proportion of incoming messages get sent to Trash?

(c) What is the probability that a message is trash given that it was sent to Trash?

3. A crushed drill core sample contains 20 gold nuggets. Assume the nuggets are randomly and independently distributed within the sample. A one– fifth part of the sample will be taken for further assay.
(a) What is the expected number of nuggets in the one–fifth part?

(b) Consider a binomial distribution model for the number of nuggets in the one–fifth part of the sample and give values for n and p.

(c) What is the probability of 2 nuggets in the one–fifth part?

(d) What is the probability of 2 or fewer nuggets in the one–fifth part?

4. The number of loss of separation incidents (LOS) in the airspace around a busy airport has averaged 8.7 per year. New radar was installed six months ago, since when there have been 8 LOS. Assume LOS occur randomly and independently. Suppose that the average annual rate of 8.7 LOS per year has not changed.
(a) What is the average rate of LOS per six months?

(b) Calculate the probability of 8 LOS in six months.

(c) Calculate the probability of 8 or more LOS in six months.

(d) It has been suggested that the new radar has led to an increase in LOS. Which of the two probabilities, (b) or (c), is the more relevant when considering this claim?

Answer 1

Quesion 1

here,

Pr(Navigation Faults) = 0.10
Pr(Ohther safety faults) = 0.20

Pr(Both faults) = 0.06

(b) Pr(either N or F) = Pr(N) + Pr(F) - 2 * Pr(N and F) = 0.20 + 0.10 - 2 * 0.06 = 0.18

(c) Pr(Neither N or F) = 1 - Pr(N or F) = 1 - [0.20 + 0.10 - 0.06] = 0.74

(d) Pr(N l F) = 0.06/0.20 = 0.30

(e) Pr(F l N) = 0.06/0.10= 0.60

Question 2

Pr(sent to trash even not trash) = 0.04

Pr(not sent to trash even if it is not trash) = 0.10

Pr(trash message) = 0.30

(a) Here the tree diagramis

(b) Pr(Sent to trash) = Pr(Trash) * Pr(Sent to trash) + Pr(not trash) * Pr(sent to trash)

= 0.30 * (1 - 0.10) + 0.70 * 0.04 = 0.298

(c) Pr(Trash l Not sent to trash)

Pr(Not sent to trash) = 1 - 0.298 = 0.702

Pr(Trash l Not sent to trash) = (0.30 * 0.10)/0.702 = 0.0427

Question 3

Here expected number of nuggets in one fifth part = 20/5 = 4

(b) Here n = 20 and p = 1/5 = 0.20

(c) Pr(X = 2) = ⁵C₂ (0.20)² (0.80)³ = 0.2048

(d) Pr(X <= 2) = Pr(X = 0) + Pr(X = 1) + Pr(X = 2) = 0.3277 + 0.4096 + 0.2048 = 0.9421

Question 4

here average number of loss of seperation incidents in the airspace= 8.7per year

(a) here the average rate of LOS per six months= 8.7/2 = 4.35

(b) Here as the distribution is Poisson Distribution where x is the number of LOS in 6 months.

Pr(x= 8) = e^-4.35 (4.35)⁸ /8! = 0.0410

(c) here

Pr(x > = 8) = 1 - Pr(x <8) = 1 - POISSON (7 ; 4.35 ; true) = 1 - 0.9253 = 0.0747

(d) Here probability c is more relevent here as the probability c only include the probability of 8 claims not more than that. So,here as the probability is not less than 0.05, so we will reject the claim

0.20 0.10 0.06

0.20 0.10 0.06