2024 HSC Mathematics Extension 1
Question 12(c):
Binomial distribution
A charity employs a worker to collect donations. There is a chance that when the charity worker talks to someone a donation is made to the charity.
Each day the charity worker must talk to exactly people.
Use the standard normal distribution and the normal probability table provided to approximate the probability that, on a particular day, at least of the people talked to made a donation.
How to recognise this question
Question type: binomial distribution
- The command and notation point to binomial distribution.
- The requested response is a worked solution.
How to handle it: identify the random variable and its parameters before using the matching probability formula.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Model with normal approximation: , .
- "At least " means at least donations, so approximate .
- Standardise: .
- From the standard normal table, .
- Hence (about ).
- NESA 12(c): 3 marks correct; 2 marks for the -score; 1 mark for a relevant standard deviation.
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
Let . Using a normal approximation (without continuity correction), approximate . Take .
Question 2
With , , approximate using and .
Question 3
For , , find and the -score for (use ), then approximate given .