Zadanie 10c, Paper 2, listopad 2022, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2022, Paper 2 (zadania wzorowane), pytanie 10(c). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
The duration T, in minutes, of a calibration cycle follows a normal distribution with mean 50 and standard deviation 8. Durations of independently selected cycles are independent.
Polecenie
A large collection contains only cycles longer than 48 minutes. (i) Find the probability that a randomly chosen cycle from this collection lasts less than 62 minutes. (ii) Twelve cycles are independently selected from the collection. Find the probability that exactly seven last less than 62 minutes.
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Odpowiedź z klucza
(i) q=P(48<T<62)/P(T>48)=0.5318991244/0.5987063257=0.8884140715≈0.888. (ii) Y~Bin(12,q); binomPDF(12,q,7)=C(12,7)q⁷(1−q)⁵=0.005985223059≈0.00599.
Schemat punktowania
(i) M1:conditional probability in this context. A1:correct ratio. A1:numerator 0.531899…. A1:q=0.888414…. (ii) M1:binomial model. A1:correct parameters 12,q and target 7. A1:0.00599. Accept correct follow-through from the conditional probability. The conditional-probability M1 also accepts the reversed conditional stated in context, but not merely P(A|B). For the next numerical A1 both correct numerator and denominator must be present or implied. In (ii), the parameter A1 is for n=12 and the candidate’s conditional p; it does not additionally require target 7. Accept the answer obtained from the correctly rounded 3sf q.
Komentarz. Paper 2: a graphic display calculator and AA HL formula booklet are allowed. Give three significant figures unless otherwise stated, and show technology inputs or mathematical setup.