9758/2019/P2/Q11

In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters.

Arif is making models of hydrocarbon molecules. Hydrocarbons are chemical compounds made from carbon atoms and hydrogen atoms. Arif has a bag containing a large number of white balls to represent the carbon atoms, and a bag containing a large number of black balls to represent the hydrogen atoms. The masses of the white balls have the distribution $\mathrm{N}\left(110,4^{2}\right)$ and the masses of the black balls have the distribution $\mathrm{N}\left(55,2^{2}\right)$. The units for mass are grams.

(i) Find the probability that the total mass of 4 randomly chosen white balls is more than 425 grams.

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(ii) Find the probability that the total mass of a randomly chosen white ball and a randomly chosen black ball is between 161 and 175 grams.

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(iii) The probability that 2 randomly chosen white balls and 3 randomly chosen black balls have total mass less than $M$ grams is $0.271$. Find the value of $M$.

[4]

Arif also has a bag containing a large number of connecting rods to fix the balls together. The masses of the connecting rods, in grams, have the distribution $\mathrm{N}\left(20,0.9^{2}\right)$. In order to make models of methane (a hydrocarbon), Arif has to drill 1 hole in each black ball, and 4 holes in each white ball, for the connecting rods to fit in. This reduces the mass of each black ball by $10 \%$ and reduces the mass of each white ball by $30 \%$.

A methane molecule consists of 1 carbon atom and 4 hydrogen atoms. Arif makes a model of a methane molecule using 4 black balls, 1 white ball and 4 connecting rods (see diagram). The balls and connecting rods are all chosen at random.

(iv) Find the probability that the mass of Arif’s model is more than 350 grams.

[4]