Marinada '25

[lang=hr] Matej kao odgovorni mentor na kampu skuplja zadatke za natjecanje RELI. Matej je jako umoran i jako rijetko budan. Naime, on bude budan $1$ minutu nakon čega spava $10$ minuta. Ovo se periodički ponavlja. Svaki put kada se probudi se s jednakom vjerojatnošću može dogoditi da je jedan papirić sa zadatkom pred njim nestalo ili da se jedan papirić sa zadatkom pred njim pojavio. Kada je počeo sastavljati RELI, vidio je samo jedan zadatak. Odredite vjerojatnost da se tokom $11$ sati ovog procesa neće ni jednom trenutku dogoditi da je broj zadataka koje Matej prilikom buđenja ugleda bude jednak $0$. Kad se zadnji put probudi, Matej će sigurno vidjeti sva $32$ zadatka i onda će moći konačno spavati u miru i pripremati MBL. [/lang] [lang=en] Matej, as a responsible mentor at the camp, is collecting problems for the RELI competition. Matej is very tired and rarely awake. Namely, he stays awake for $1$ minute, after which he sleeps for $10$ minutes. This cycle repeats periodically. Each time he wakes up, with equal probability, one problem sheet in front of him may disappear or a new one may appear. When he started preparing RELI, he saw only $1$ problem. Determine the probability that during $11$ hours of this process, the number of problems Matej sees upon waking is never equal to $0$. When he wakes up for the last time, Matej will certainly see all $32$ problems, and then he will finally be able to sleep peacefully and prepare for MBL. [/lang]