A tray has 12 eggs, of which 4 are rotten. Three eggs are taken out at random, one by one, without replacement. The probability that none of the three eggs is rotten is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{14}{55}\)
There are 8 good eggs. Each time a good egg is taken out, there is one good egg fewer and one egg fewer in all. By the multiplication theorem:
\( \displaystyle P = \frac{8}{12}\times\frac{7}{11}\times\frac{6}{10} \)
\( \displaystyle = \frac{336}{1320} = \frac{14}{55} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{8}{27}\) — Incorrect. This is \(\left(\frac{8}{12}\right)^3\), which treats the draws as made with replacement.
Option (3) \(\displaystyle\frac{28}{55}\) — Incorrect. This is the probability that exactly one of the three eggs is rotten.
Option (4) \(\displaystyle\frac{1}{55}\) — Incorrect. This is \(\frac{4}{12}\times\frac{3}{11}\times\frac{2}{10}\), the probability that all three eggs are rotten.