Zadanie 9e, Paper 2, maj 2023, TZ2, Matematyka AA IB SL
Wzorowane na: Matematyka AA IB SL — maj 2023, TZ2, 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 shuffled set contains n cards, of which 8 are marked and the rest unmarked. Cards are drawn without replacement. In parts 9b–9e, n=20. For a game, three cards are drawn: exactly two marked cards earn 18 points, all three marked earn 45 points, and any other outcome earns zero. All three cards are returned and the set reshuffled before each new game.
Polecenie
Find the smallest number k of games for which the expected total score exceeds 160.
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Exactly two marked cards have probability 3(8/20)(7/19)(12/18) = 28/95 (three possible positions for the unmarked card). Three marked have probability (8/20)(7/19)(6/18) = 14/285. Expected points per game are 18(28/95) + 45(14/285) = 714/95 ≈ 7.51579. Hence k(714/95)>160 gives k>7600/357≈21.2885, so the least integer is 22.
Schemat punktowania
A1:probability 28/95 for exactly two marked. A1:probability 14/285 for three marked. M1:form the weighted expectation. A1:714/95 per game. M1:set up an equation or inequality comparing expected total score with 160. A1:k=22, using an admissible expectation. The first two accuracy marks are independent.
Komentarz. Paper 2: calculator and AA SL formula booklet allowed. Retain intermediate precision; give three significant figures unless an exact answer is requested.