Review practice · Section 1 of 2

The times taken by two runners A and B to run 400 metre races are independent and normally distributed with means 45.0 seconds and 45.2 seconds and standard deviation 0.5 seconds and 0.8 seconds respectively. The two runners are to compete in a 400 metre race for which there is a track record of 44.5 seconds.

(i) Calculate the probability of runner A breaking the track record. Hence, show that the probability of runner B breaking the record is greater than that of runner A.3 marks

(ii) Find the probability of runner A beating runner B.3 marks

(iii) Find the probability that the total sum of four randomly chosen times of A is more than 4 times a randomly chosen time of B.5 marks

Durians cost 7 dollars per kg; a durian’s mass is normal with mean 1.5 kg and standard deviation 0.2 kg. Mangoes are sold in bags of 5 at 2 dollars per kg. One mango’s mass is normal with mean 0.4 kg and standard deviation 0.05 kg. Fruit masses are independent; empty bag mass is negligible.

(i) Find the probability that a random bag of mangoes costs more than 4.20 dollars.3 marks

(ii) Find the probability that both a random durian costs more than 10 dollars and a random mango bag more than 4.20 dollars.3 marks

(iii) Find the probability that 2 durians and 3 mango bags cost less than 33.50 dollars in total.5 marks

Oil volumes in Small, Regular and Large bottles are normal.

Bottle	Mean (ml)	Variance (ml²)
Small	610	25
Regular	1050	$\sigma^2$
Large	1200	50

(i) Explain why Small volume can be modelled as normal although oil volume cannot be negative.1 mark

(ii) Three Small bottles are randomly chosen. Find the probability that one contains more than 615 ml and the other two less than 602 ml each.3 marks

(iii) Find σ if 80% of Regular bottles contain between 1040 and 1060 ml.3 marks

(iv) Find the probability that two random Large bottles contain more than four times the volume in one random Small bottle.3 marks

The source also prints “The mass of an empty bag is assumed to be negligible”; this unrelated bag note has no role in the bottle-volume model.

Independent call durations to City A are normal with mean 8 minutes and standard deviation 2.5 minutes; to City B, mean 10 and standard deviation 3.2.

(i) Find the probability that two random City A call durations in total differ from three times a third City A duration by more than 3 minutes.3 marks

Calls cost 22 cents per minute to A and 30 cents per minute to B, plus a 10 cent handling fee per call. Ann makes two calls to A and one to B.

(ii) Find her expected total cost.1 mark

(iii) Show that the probability her cost exceeds 8.50 dollars is 0.0870 to 3 significant figures.2 marks

On Crabby Island, a kelong rears Hairy Crabs with mass $H\sim N(\mu,\sigma^2)$ kg. Given $P(H<1.5)=P(H>3.5)=0.33$, state μ.1 mark

(i) Show that σ=2.27.2 marks

(ii) For 4 random Hairy Crabs, find the probability that two weigh between 1.5 and 3.5 kg each and the other two exceed 3.5 kg each.3 marks

Flower Crab mass has an independent normal distribution with mean 3.2 kg and standard deviation 1.3 kg. Find the largest a such that the probability the average mass of 2 Hairy and 3 Flower Crabs exceeds a kg is more than 0.6.3 marks

Peter Piper guarantees every pickle is at least 8 cm long; his bottles hold pickles of at most 10.5 cm. Pickle length is normal with mean 9 cm and standard deviation 0.8 cm.

(i) Find the probability a random pickle must be discarded because(a) it is below the guaranteed length 1 mark or (b) it does not fit the bottle.1 mark

(ii) He wants to discard no more than 5% of pickles. Comment on whether this is possible.1 mark

(iii) Find the probability that both of two randomly selected pickles fail the guarantee, given that their total length is below 16 cm.3 marks