Zadanie 1h, Paper 3, maj 2026, TZ2, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2026, TZ2, Paper 3 (zadania wzorowane), pytanie 1(h). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
Darmowe konto pozwoli wrócić do niego później.
Zadanie
Wspólne informacje do wiązki zadań
An energy-buffer simulation has integer charge levels 0,1,…,w. While the charge is strictly between 0 and w, each independent tick increases it by one with probability p and decreases it by one with probability q=1−p, where 0<p<1 and p≠q. The simulation stops as soon as charge reaches 0 (shutdown) or w (full charge). Let Rₙ be the probability of reaching w before 0, starting at charge n. Thus R₀=0, R_w=1, and Rn = qRn−1 + pRn+1 for 1 ≤ n ≤ w − 1.
Polecenie
The upward-tick probability is now 0.498. A run starts with m charge units and stops successfully when it has accumulated 12 additional units, so capacity is w=m+12. Find the smallest positive integer m giving at least an 85% chance of success.
Korepetytor
Nie wiesz, jak ruszyć? Korepetytor zna to zadanie i poprowadzi Cię krok po kroku, bez gotowca.
3 rozmowy · bez karty · Zaloguj się
Sprawdź się
Jak poszło?
Pokaż rozwiązanieUkryj rozwiązanie
Odpowiedź z klucza
Let ρ=0.502/0.498. The required inequality is (1 − ρm)/(1 − ρm+12) ≥ 0.85. Numerical solving gives m≥105.7783462… . At m=105 the probability is 0.8493991910… and at 106 it is 0.8501695406… . Thus the least integer is 106. For ρ>1 this probability increases with m, since it can be written (1−ρ^(−m))/(ρ^12−ρ^(−m)).
Schemat punktowania
A1: correct probability inequality with capacity m+12; also accept the equivalent recurrence form qR_(m-1)+pR_(m+1)>=0.85, with both R terms evaluated using the same capacity m+12. M1: solve using a graph, equation or table. A1: threshold 105.778… or correct probabilities bracketing it. A1: minimum integer 106. An equality or strict inequality is acceptable while locating the threshold, provided the final integer satisfies the requested inclusive condition.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Give exact values or three significant figures unless requested otherwise; proofs and numerical solutions require supporting work.