Binomial distribution — Bikes and free weights

Question

A gym has bikes and free-weight stations. Each bike is in use with probability , and each free-weight station is free (unused) with probability , independently.

(i) Find an expression for the probability that, at a particular time, exactly of the bikes are in use.

(ii) Hence find an expression for the probability that exactly of the bikes are in use and all free-weight stations are free.

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.

Step by step

Set up the solution: identify the random variable and its parameters before using the matching probability formula.(i) Binomial: .
Finish the calculation, then check that the result meets the question’s conditions.(ii) Independently, all free-weight stations free: .
State the final answer clearly in the form the question requested.Combined (product of independent events): .

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

Verification: Full (i)–(ii): binomial for group 1, then multiply by independent free probability for group 2.

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

← Back to 2023 HSC Extension 1 paper Question 12(c)