Zadanie 11f, Paper 2, maj 2023, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2023, TZ1, Paper 2 (zadania wzorowane), pytanie 11(f). 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 continuous random variable X has density for , and zero otherwise. It models a robotic alignment error in millimetres. Independent trials are successful when X<0.6. For display purposes, trial outcomes can be placed in numbered slots around a circle; the last slot is adjacent to the first. Rotations of slot labels count as different placements.
Polecenie
Consider n≥4 trials with exactly three successes, placed in n numbered circular slots in trial order. (i) Give the number of placements in which all three success slots are consecutive around the circle. (ii) Find the greatest n for which the conditional probability of this event is greater than 0.06.
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Odpowiedź z klucza
(i) There are n distinct length-three circular blocks. (ii) Conditional on exactly three successes, every three-slot subset has the same probability. There are C(n,3) subsets, so the desired probability is n/C(n,3)=6/[(n−1)(n−2)]. Require (n−1)(n−2)<100. The positive boundary is (3+√401)/2≈11.5125; hence the greatest integer is 11. At 11 the probability is 1/15≈0.0667; at 12 it is 3/55≈0.0545.
Schemat punktowania
(i) A1:n. (ii) A1:C(n,3) equally likely subsets. A1:correct conditional probability ratio. M1:solve the strict inequality or boundary equation. A1:correct boundary or checks at 11 and 12. A1:greatest n=11.
Komentarz. Paper 2: calculator and AA HL formula booklet allowed. Give exact values where requested; otherwise retain intermediate precision and use three significant figures, except the specified six-significant-figure approximations.