2024 HSC Mathematics Extension 1
Question 7:
Binomial distribution
A driver's knowledge test contains multiple-choice questions, each with options. An applicant must get at least correct to pass.
If an applicant correctly answers the first questions and randomly guesses the last questions, what is the probability that the applicant will pass the test?
How to recognise this question
Question type: binomial distribution
- The command and notation point to binomial distribution.
- The requested response is a multiple choice.
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
- With already correct, the applicant needs at least of the last guesses correct to reach .
- Each guess is independent with success probability (one correct option among four).
- Let . Pass iff .
- .
- .
- Total: .
- Answer: C. (NESA multiple-choice key: C.)
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 1 total mark 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
An applicant has correct already and needs of . They randomly guess remaining four-option items. Probability of at least correct is
Question 2
With five independent four-option guesses, equals
Question 3
For , equals