2023 HSC Mathematics Extension 1
Question 2:
Binomial distribution
A standard six-sided die is rolled times.
Let be the proportion of the rolls with an outcome of .
Which of the following expressions is the probability that at least of the rolls have an outcome of ?
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
- At least successes in trials means the sample proportion satisfies .
- So the required probability is .
- Option B uses the theoretical , not the sample proportion threshold.
- Options C and D reverse the inequality direction.
- Answer: A. (NESA multiple-choice key: A.)
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
A coin is tossed times. Let be the proportion of heads. The probability of at least heads is
Question 2
In independent trials, is the sample proportion of successes. equals
Question 3
A die is rolled times; is the proportion of sixes. Probability of at least sixes is