I had this question in my past paper:
Bags of sugar are packed in boxes, each box containing 20 bags. The masses of the boxes, when
empty, are normally distributed with mean 0.4 kg and standard deviation 0.01 kg. The masses of the
bags are normally distributed with mean 1.02 kg and standard deviation 0.03 kg.i) Two full boxes are chosen at random. Find the probability that they differ in mass by less than 0.02 kg
I did the calculations and I got my probability as 2 x 0.4582 however in the marking scheme they've deducted the final answer from 1. Why is that so?
Best Answer
I do not know how you arrived at your result but I would do in the following manner:
Set X= boxes' mass and Y=bags' mass.
The gross mass is
$$Z=(X+Y_1+Y_2+\dots +Y_{20})\sim N(20.8;0.0181)$$
and
$$W=(Z_1-Z_2)\sim N(0;0.0362)$$
you are requested to calculate
$$\mathbb{P}[|W|<0.02]=\mathbb{P}[-0.02<W<0.02]=$$
$$=\Phi\left(\frac{0.02}{\sqrt{0.0362}} \right)-\Phi\left(\frac{-0.02}{\sqrt{0.0362}} \right)=0.5419-0.4581=0.0837$$