We sequentially crack ten eggs into a bowl. Among the eggs, two are rotten, but this only becomes apparent when they are cracked. The yolks render all previously cracked eggs unusable. In such a case, we discard the bowl and continue the process with the remaining eggs. On average, what fraction of the good eggs will be wasted this way?
Clearly, only the eggs that break after the second drop are not wasted. In the diagram, all possible locations of the second drop are shown in black.

In case I, the first drop can only occur at one (the first) location, and the number of good eggs wasted is 0. In case II, the first drop can occur at two locations, and 1 good egg is wasted. In case III, the first drop can be chosen from 3 locations, and 2 good eggs are always wasted, and so on. In case IX, the first drop can occur at 9 locations, and 8 good eggs are wasted.

Thus, the number of good eggs wasted is:

$$
1 \cdot 0 + 2 \cdot 1 + 3 \cdot 2 + \cdots + 9 \cdot 8 = 240
$$

eggs. There were always 8 good eggs, totaling $(1 + 2 + \cdots + 9) \cdot 8 = 360$ eggs.

Since 240 out of 360 are wasted, on average, $\frac{240}{360} = \frac{2}{3}$ of the good eggs are wasted. Schlotter Diana (Budapest, Veres Péter Gymn., 11th grade)

| I. | 8 good eggs not wasted | 0 good eggs wasted |
| :---: | :---: | :---: |
| II. | 7 good eggs | 1 good egg |
| III. | 6 good eggs | 2 good eggs |
| IV. | 5 good eggs | 3 good eggs |
| V. | 4 good eggs | 4 good eggs |
| VI. | 3 good eggs | 5 good eggs |
| VII. | 2 good eggs | 6 good eggs |
| VIII. | 1 good egg | 7 good eggs |
| IX. | 0 good eggs | 8 good eggs |