2023 HSC Mathematics Extension 1
Question 12(c):
Binomial distribution
A gym has pieces of equipment: treadmills and rowing machines.
On average, each treadmill is used of the time and each rowing machine is used of the time.
(i) Find an expression for the probability that, at a particular time, exactly of the treadmills are in use.
(ii) Find an expression for the probability that, at a particular time, exactly of the treadmills are in use and no rowing machines are in use.
How to recognise this question
Question type: binomial distribution
- The command and notation point to binomial distribution.
- The word “hence” means the later part is intended to reuse the earlier result.
How to handle it: identify the random variable and its parameters before using the matching probability formula.
Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.
Step-by-step answer
- (i) Model treadmill use as independent Bernoulli trials with .
- Exactly of in use: .
- NESA (i): 2 marks correct; 1 for some binomial structure.
- (ii) Independently, each rower unused with probability , so all unused: .
- Combined: .
- NESA (ii): 1 mark for the correct product expression.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
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.
Question 2
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.
Question 3
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.