The box-and-whisker plots show the distribution of the masses (in grams) of eggs sold in Shop A and Shop B.

1. Relate masses above the upper quartile to one quarter of the total number of eggs.

For Shop A, $Q_3 = 59$ g.

So the eggs with masses more than $59$ g make up $25\%$ of the total.

$25\% \text{ of total} = 25$

$\text{Total} = 25 \div 0.25 = 100$

1. Compare the averages using the medians from the box-and-whisker plots.

For a box-and-whisker plot, the average is usually compared using the median.

Shop A median $=54$ g.

Shop B median $=50$ g.

Since $54>50$, Shop A has the higher average mass.

2. Compare the distributions using a measure of spread from the box-and-whisker plots.

Use the interquartile range to compare spread.

Shop A:
$\text{IQR}=Q_3-Q_1=59-48=11$ g.

Shop B:
$\text{IQR}=Q_3-Q_1=58-46=12$ g.

Since $12>11$, Shop B has a slightly greater spread in the middle $50\%$ of the masses.

1. Find the probability that the selected egg is cracked.

There are $4$ cracked eggs out of $80$ eggs.

$P(\text{cracked})=\frac{4}{80}=\frac{1}{20}$

1. Find the probability that the first egg selected is cracked.

There are $4$ cracked eggs out of $80$ eggs.

$P(\text{first cracked})=\frac{4}{80}$

2. Account for no replacement and multiply the probabilities.

After selecting one cracked egg, there are $3$ cracked eggs left out of $79$ eggs.

$P(\text{both cracked})=\frac{4}{80}\times\frac{3}{79}=\frac{12}{6320}=\frac{3}{1580}$

1. Find the probability that none of the three selected eggs is cracked.

There are $80-4=76$ uncracked eggs.

$P(\text{none cracked})=\frac{76}{80}\times\frac{75}{79}\times\frac{74}{78}$

2. Use the complement to find the probability that at least one egg is cracked.

$P(\text{at least one cracked})=1-\frac{76}{80}\times\frac{75}{79}\times\frac{74}{78}$

$=1-\frac{5615}{7900}$

$=\frac{2285}{7900}=\frac{229}{790}$