Zadanie 9d, Paper 2, maj 2026, TZ1, Matematyka AA IB SL
Wzorowane na: Matematyka AA IB SL — maj 2026, TZ1, Paper 2 (zadania wzorowane), pytanie 9(d). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
A festival game uses a bag containing red, blue and gold markers. There are exactly 18 gold markers and equal numbers of red and blue markers. A gold marker is drawn with probability 0.45. Each play consists of 12 independent draws with replacement, recording the colour each time.
Polecenie
One play costs 6 credits. At least seven red markers win 50 credits, at least seven blue markers win B credits (a positive integer), and at least seven gold markers win 15 credits. Otherwise the prize is zero. These prize events are mutually exclusive. Let X be the net gain after paying the fee. The outcome table gives these net gains and four-significant-figure probabilities: red prize, gain 44, probability 0.02419; blue prize, gain B−6, probability 0.02419; gold prize, gain A, probability 0.2607; no prize, gain −6, probability p. If two net gains coincide, their outcome probabilities are combined in the distribution of X.
(i) Use the tabulated probabilities to find p. (ii) Find the smallest integer B giving positive expected net gain.
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Odpowiedź z klucza
(i) p = 1 − 2(0.02419) − 0.2607 = 0.69092. (ii) E(X) = 44(0.02419) + (B − 6)(0.02419) + 9(0.2607) − 6(0.69092) = 0.02419B − 0.88000. Positivity requires B>36.37866887… . The smallest integer is B=37. Equivalently compare the expected prize 50(0.02419)+B(0.02419)+15(0.2607) with the fee 6.
Schemat punktowania
(i) M1: probabilities sum to 1. A1: p=0.69092 or a correctly rounded value. (ii) M1: form expected net gain, or expected prize less the fee. A1: correct weighted expression/inequality. A1: smallest integer B=37.
Komentarz. Paper 2: calculator and AA SL formula booklet allowed. Give exact results or three significant figures unless specified otherwise; show the required mathematical or technology inputs.