Practice section 6 of 7

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]

(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]

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

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]

(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]

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

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

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]

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]

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

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]

(i) Show that σ=2.27. [2]

(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]

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]

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] or (b) it does not fit the bottle. [1]

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

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

Girl heights are normal with mean 155 cm and standard deviation 10 cm. Find

(i) the symmetric interval about the mean containing 40% of heights, [3]

(ii) the probability exactly two of four randomly chosen girls exceed 170 cm, [3]

(iii) the probability that the total height of n randomly chosen girls exceeds 150 n cm, for large n. [3]