While investigating a case, Detective Bashkovsky discovered that the key witness was the one from the Petrovs family who came home before the others on that fateful day. The investigation revealed the following facts.

1. Neighbor Marya Kuzminichna wanted to borrow some salt from the Petrovs, rang their doorbell, but no one answered. When? Who knows? It was already dark...
2. Galina Efimovna Petrova, coming home in the evening, found both children in the kitchen, and her husband on the sofa - he had a headache.
3. Husband Anatoly Ivanovich claimed that as soon as he came home, he immediately lay down on the sofa and dozed off, saw no one, heard nothing, the neighbor definitely didn't come - the doorbell would have woken him up.
4. Daughter Svetlana said that upon returning home, she immediately went to her room, knew nothing about her father, but as usual, stumbled over Dima's shoe in the hallway.
5. Dmitry didn't remember when he came home, didn't see his father, but heard Svetka swearing about the shoe.

"Hmm," thought Bashkovsky. "What is the probability that Dmitry returned home before his father?"
Regarding Dmitry, Svetlana, and Anatoly, there is no certainty; it is only known that all three arrived home between the visit of Marya Kuzminichna and the arrival of Galina Efimovna - these two events delimit the time period of interest to Bashkovitsky. We will assume that the probability of a family member arriving home within a certain time interval is proportional to the length of that interval.

Let's introduce the notation. Let the time from Marya Kuzminichna's visit to Galina Efimovna's arrival be $m$, the time until Anatoly's arrival be $x$, the time until Dmitry's arrival be $y$, and the time until Svetlana's arrival be $z$.

From the witness statements, it follows that $0 < x < m$, $0 < y < z < m$. In the coordinate system $Oxyz$, we obtain a triangular prism $F$ with vertices $(0,0,0)$, $(0,0,m)$, $(0,m,m)$, $(m,0,0)$, $(m,0,m)$, and $(m,m,m)$ (see figure). Under these conditions, we need to find the probability of the event $y < x$. Inside the prism $F$, this inequality defines a quadrilateral pyramid $G$ with vertices $(0,0,0)$, $(m,0,0)$, $(m,0,m)$, $(0,0,m)$, and $(m,m,m)$.

![](https://cdn.mathpix.com/cropped/2024_05_08_293b4772a418c01a5915g-19.jpg?height=520&width=504&top_left_y=2204&top_left_x=777)

The desired probability is $\frac{V_{G}}{V_{F}} = \frac{m^{3}}{3} : \frac{m^{3}}{2} = \frac{2}{3}$.

## Answer

## Problem