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(a) The number N of workplace health and safety incidents per month that occurs at a large oil refinery

is expected to agree with the following table:

n

0-2 3 4 5

Pr(N = n) 0.15 0.27 k 0.12 0.15

≥6

i) (1 mark) Find the value of k which makes this a valid probability distribution.

ii) (2 marks) Over the last 12 months, there are two months in which there have been at least 6

workplace health and safety incidents per month. Use the binomial distribution to estimate

the probability of this occurring.

(b) A certain car make and model is under warranty for the first 10 years after being purchased, so will

only potentially require the consumer to pay for significant mechanical repairs after this 10-year

period. Hence, the number of years T after being purchased until this car requires the consumer to

pay for significant mechanical repairs is distributed according to the following probability density

function (pdf):

f(t) =

20

12

10 < t < 20,

otherwise.

i) (2 marks) What is the probability that a consumer will need to pay for significant mechanical

repairs less than 18 years after the car is purchased? (i.e. less than 8 years after the warranty

period expires?)

ii) (2 marks) It can be shown from the above equation that the expected time until the consumer

would need to pay for significant mechanical repairs is E(T) 13.86. Use this information to

determine the variance Var(T) of the variable T./n(c) The quality of a manufactured gold component is assessed for both its purity and its conductivity

prior to being shipped. Investigations have shown that the probability that the gold component has

sufficient purity to be shipped is 90%, and the probability that the gold component has sufficient

conductivity to be shipped is 80%. The probability that the gold component is both sufficiently

pure and sufficiently conductive to be shipped is 75%.

i) (2 marks) Calculate the probability that the gold component is neither sufficiently pure nor

sufficiently conductive to be shipped.

ii) (1 mark) Are the two events - of the gold component being sufficiently pure, and of the gold

being sufficiently conductive mutually exclusive or independent or neither? Explain your

answer.

iii) (1 mark) If a gold component is deemed to be sufficiently conductive to be shipped, what is

the probability that it is sufficiently pure to be shipped?

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