2022 HSC Mathematics Extension 1
Question 12(e):
Binomial distribution
A game consists of randomly selecting 4 balls from a bag. After each ball is selected it is replaced. The bag contains 3 red balls and 7 green balls. For each red ball selected, 10 points are earned and for each green ball selected, 5 points are deducted.
What is the expected score in the game?
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
- Each draw: , .
- Expected score on one draw: .
- By linearity (with replacement, identical trials): expected total for 4 draws is .
- Equivalently: expected red count , green ; score .
- NESA: 2 marks correct; 1 mark for using a binomial/expectation idea.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 2 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
Same bag, 4 draws with replacement; red , green . Expected score?
Question 2
Bag: 2 red, 8 green; 5 draws with replacement; red , green . Expected score?
Question 3
With 3 red and 7 green, points / , show the one-draw expectation is .