Zadanie 9e, Paper 2, maj 2026, TZ1, Matematyka AA IB SL
Wzorowane na: Matematyka AA IB SL — maj 2026, TZ1, Paper 2 (zadania wzorowane), pytanie 9(e). 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.
Plays are independent. Find the least number of plays for which the probability of winning at least one prize exceeds 0.99.
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Odpowiedź z klucza
No prize in one play has probability p=0.69092. Thus P(at least one prize in n plays) = 1 − pn. Solve 1 − 0.69092n > 0.99, giving n > ln(0.01)/ln(0.69092) = 12.45545341…. Hence the least integer is n=13. Use of the unrounded binomial probabilities also gives 13.
Schemat punktowania
M1: identify the one-play no-prize probability p. A1: expression 1−p^n. M1: solve the inequality or compare successive integer trials. A1: 13 plays.
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.