Olive Oil
In an olive oil factory, bottles are either good (G) or bad (B). If a bottle is G, the next is G with probability \(\frac{2}{5}\); if B, the next is B with probability \(\frac{3}{5}\). Given that the first bottle is B, what is the expected number of bottles until another B appears?
Answer
Let
\[
E = \text{expected number of bottles until the next B (starting from B)}
\]
and
\[
E_G = \text{expected number until a B appears (starting from G)}.
\]
Condition on the quality of the next bottle:
\[
E = \frac{3}{5}\cdot 1 + \frac{2}{5}\Bigl(1 + E_G\Bigr).
\]
Similarly, starting from G:
\[
E_G = \frac{3}{5}\cdot 1 + \frac{2}{5}\Bigl(1 + E_G\Bigr)
\quad\Longrightarrow\quad \frac{3}{5}E_G = 1,
\quad\Longrightarrow\quad E_G = \frac{5}{3}.
\]
so basically a geometric random variable (shortcut). Substitute back into the equation for \(E\):
\[
E = \frac{3}{5} + \frac{2}{5}\left(1 + \frac{5}{3}\right)
= \frac{3}{5} + \frac{2}{5}\cdot\frac{8}{3}
= \frac{5}{3}.
\]
Thus, the expected number of bottles until another B appears is \(\boxed{\frac{5}{3}}\).