Zadanie 1a, Paper 3, maj 2025, TZ1, Matematyka AA IB HL

A polynomial can encode a finite probability distribution: for a nonnegative integer-valued random variable U, define $G_U(t)=\sum_u P(U=u)t^u$. Two independent random selectors each return one of 1, 2, 3, 4 with equal probability. Let M be the smaller of the two outputs. Its probabilities for m = 1, 2, 3, 4 are respectively 7/16, 5/16, 3/16, 1/16, so G_M(t) = (7t + 5t² + 3t³ + t⁴)/16. In later parts, X counts experts in a two-person panel chosen uniformly from seven people, three experts and four trainees. Independently of that selection, two sensors attempt to register a pulse; their independent detection probabilities are 0.4 and p. Let Y count detections. The coefficient of t² in G_Y(t) is 0.24. For Z = X + Y, you may use G_Z(t) = G_X(t)G_Y(t).