15. Luca and Carlo are playing bingo against each other (with no other opponents). Each has a card with 15 numbers; the two cards do not have any numbers in common. What is the probability that the game will end exactly on the 89th number drawn?
(A) $9 / 178$
(B) $5 / 178$
(C) $5 / 89$
(D) $1 / 18$
(E) $1 / 36$
Problem 15. The correct answer is (C).

Imagining to complete the extraction of the 90 numbers, emptying the entire urn, the event $E$ indicated occurs if and only if the 90th number drawn belongs to one of the two cards while the 89th belongs to the other. Indicating, for $1 \leqslant n \leqslant 90$, with $C_{n}$ and $L_{n}$ the events

$$
C_{n}: \text { the } n \text {-th draw is in Carlo's card } \quad L_{n} \text { : the } n \text {-th draw is in Luca's card, }
$$

it will therefore be that $E=\left(C_{90} \cap L_{89}\right) \cup\left(L_{90} \cap C_{89}\right)$ and consequently:

$$
\mathcal{P}(E)=\mathcal{P}\left(C_{90}\right) \cdot \mathcal{P}\left(L_{89} \mid C_{90}\right)+\mathcal{P}\left(L_{90}\right) \cdot \mathcal{P}\left(C_{89} \mid L_{90}\right)=\frac{15}{90} \cdot \frac{15}{89}+\frac{15}{90} \cdot \frac{15}{89}=\frac{5}{89}
$$