14. Anya, Borya, and Vasya decided to go to the "Fir Tree". They agreed to meet at the bus stop, but they don't know when each of them will arrive. Each of them can arrive at a random time between 15:00 and 16:00. Vasya is the most patient of all: if he arrives and there is neither Anya nor Borya at the stop, he will wait for 15 minutes, and if he doesn't wait for anyone, he will go to the "Fir Tree" alone. Borya is less patient: he will wait only 10 minutes. Anya is the most impatient: she won't wait at all. However, if Borya and Vasya meet, they will wait for Anya until 16:00. What is the probability that they will all go to the "Fir Tree" together?
# Solution.

Since Anya will not wait at all, the boys will only go to the "Fir Tree" together if Anya arrives last. Clearly, these events are independent, so

$$
\begin{gathered}
\mathrm{P}(\{\text { all three will go to the "Fir Tree" together }\})= \\
=\mathrm{P}(\{\text { Anya will arrive last }\}) \mathrm{P}(\{\text { Borya and Vasya will meet }\})
\end{gathered}
$$

From combinatorial considerations:

$$
\mathrm{P}(\{\text { Anya will arrive last }\})=\frac{2(\text { number of suitable options })}{3!(\text { number of permutations })}=\frac{1}{3} .
$$

The second probability is found geometrically (see figure):

![](https://cdn.mathpix.com/cropped/2024_05_06_dae93645f03c90e747e3g-12.jpg?height=711&width=785&top_left_y=136&top_left_x=681)

The figure outlined in black is the set of arrival times for both, during which they will meet. The probability will be calculated as the ratio of the times of arrival that satisfy the condition of meeting to the set of possible arrivals. Therefore, the area of the highlighted part must be divided by the area of the square \(12 \times 12\), if the area is calculated by cells.

$$
\mathrm{P}(\{\text { Borya and Vasya will meet }\})=\frac{12^{2}-\frac{9^{2}}{2}-\frac{10^{2}}{2}}{12^{2}} \approx 0.371527778
$$

Therefore,

$$
\begin{gathered}
\mathrm{P}(\{\text { all three will go to the "Fir Tree" together }\}) \approx \\
\approx \frac{0.371527778}{3} \approx 0.123842593 \approx 0.124 .
\end{gathered}
$$

Answer: 0.124.