Binomial distribution — Cross-trainers and rowers

Question

A gym has cross-trainers and rowing machines. Each cross-trainer is used with probability and each rower is unused with probability , independently.

(i) Find an expression for the probability that exactly of the cross-trainers are in use.

(ii) Hence find an expression for the probability that exactly of the cross-trainers are in use and none of the rowers are in use.

This practice question was inspired by Question 12(c) in the 2023 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: identify the random variable and its parameters before using the matching probability formula.(i) .
Finish the calculation, then check that the result meets the question’s conditions.(ii) None of rowers in use: .
State the final answer clearly in the form the question requested.Combined: .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Same two-group independence structure as the 2023 paper.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.