If X is the random variable representing the small bowl and Y the random variable representing the large bowl, then \(\displaystyle{Y}-{X}\) represents the difference between the large and small bowl.

a) The expected value changes the same as the random variables:

\(\displaystyle\mu-\mu_{{Y}}-\mu_{{X}}={2.5}-{1.5}={1}\)

b) The variance changes by the square of the coefficients and the standard deviation is the square root of the variance:

\(\displaystyle\sigma=\sqrt{{{\sigma_{{Y}}^{{2}}}+{\sigma_{{X}}^{{2}}}}}=\sqrt{{{0.4}^{{2}}+{0.3}^{{2}}}}={0.5}\)

c) Determine the z-score:

\(\displaystyle{z}={\frac{{{0}-{1}}}{{{0.5}}}}=-{2}\)

Look up the corresponding probability in the table

\(\displaystyle{P}{\left({Y}-{X}{<}{0}\right)}={0.0228}={2.28}\%\)

If X is the random variable representing the small bowl and Y the random variable representing the large bowl, then Y+X represents the total amount of the large and small bowl.

d) The expected value changes the same as the random variables:

\(\displaystyle\mu=\mu_{{Y}}+\mu_{{X}}={2.5}+{1.5}={4}\)

The variance changes by the square of the coefficients and the standard deviation is the square root of the variance:

\(\displaystyle\sigma=\sqrt{{{\sigma_{{Y}}^{{2}}}+{\sigma_{{X}}^{{2}}}}}=\sqrt{{{0.4}^{{2}}+{0.3}^{{2}}}}={0.5}\)

e) Determine the z-score

\(\displaystyle{z}={\frac{{{4.5}-{4}}}{{{0.5}}}}={1}\)

Look up the corresponding probability in the table

\(\displaystyle{P}{\left({Y}+{X}{>}{4.5}\right)}={1}-{0.8413}={0.1587}={15.87}\%\)

f) If X is the random variable representing the small bowl and Y the random variable representing the large bowl and Z represents the amount in the box, then \(\displaystyle{Z}-{\left({X}+{Y}\right)}\) represents the total amount left in the box.

The expected value changes the same as the random variables:

\(\displaystyle\mu=\mu_{{Z}}-\mu_{{X}}-\mu_{{Y}}={16.3}-{1.5}-{2.5}={12.3}\)

The variance changes by the square of the coefficients and the standard deviation is the square root of the variance:

\(\displaystyle\sigma=\sqrt{{{\sigma_{{Z}}^{{2}}}+{\sigma_{{Y}}^{{2}}}+{\sigma_{{X}}^{{2}}}}}=\sqrt{{{0.2}^{{2}}+{0.4}^{{2}}+{0.3}^{{2}}}}\approx{0.54}\)