Marinada '24

[lang=hr] Konačno, došao je trenutak da Matej pogađa $p$ na Lucijinom novčiću. Uzmite sve iste pretpostavke kao u prethodna dva zadatka u ovom lancu. Da bude zanimljivije, odlučili su se kladiti u 100 eura da će Matej pogriješiti, a Luciji će platiti ovisno o kvadratu greške koji napravi. Formula koju koriste je da ukoliko je njegov pogodak $p_1$, a stvarna vjerojatnost $p_2$, dat će joj $100 * (p_1-p_2)^2$ eura. Matej želi izgubiti što manje novca, stoga ga zanima koja je vrijednost $p$ koju bi trebao pogoditi da minimizira očekivani gubitak? [/lang] [lang=en] Finally, the moment has come for Matej to guess $p$ on Lucija’s coin. Assume all the same conditions as in the previous two problems in this series. To make it more interesting, they decided to bet €100 that Matej will guess wrong, and Matej will pay Lucija based on the square of the error he makes. The formula they use is that if his guess is $p_1$ and the actual probability is $p_2$, he will pay her €$100 * (p_1 - p_2)^2$. Matej wants to lose as little money as possible, so he’s wondering what value of $p$ he should guess to minimize his expected loss? [/lang]