Binomial distribution — Treadmills and bikes

Question

A gym has treadmills and bikes. Each treadmill is used with probability and each bike is unused with probability , independently.

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

(ii) Hence find an expression for the probability that exactly of the treadmills are in use and no bikes 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) No bikes in use: all unused with probability .
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: Hence multiplies the part (i) expression by the independent all-unused factor.

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