Question 2
A binomial random variable arises from the number of successes in $n$ independent Bernoulli trials.
A context not suitable for modelling using a binomial random variable is recording the number of
Not quite. This one is binomial. Each toss is heads or not heads, there are 12 tosses, the chance of heads is the same on every toss, and one toss doesn’t affect another. In 2022, 1.7% of Queensland students chose (A).
Not quite. This one is binomial. Each person is left-handed or not, there are 100 people, and in a large population each person picked has about the same chance of being left-handed. In 2022, 11.3% of Queensland students chose (B).
Not quite. This one is binomial. Each shot is a hit or a miss, there are 20 shots, and the question tells you each shot is independent of all the others. In 2022, 10.6% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2022, 76.0% of Queensland students chose (D).
Nothing has been drawn yet, so the chance of red is $\tfrac{5}{9}$. Make a draw and watch the bag.
Each marble drawn stays out, so the chance of red changes with what came out before. On draw 2 it’s $\tfrac{4}{8}$ after a red but $\tfrac{5}{8}$ after a blue. The draws aren’t independent and the chance isn’t fixed, so this isn’t binomial.
Each marble goes back, so the chance of red is $\tfrac{5}{9}$ on every draw and no draw affects another. With replacement this would be binomial. The question’s marbles are drawn without replacement, so it isn’t.
A binomial counts successes over a set of trials. Check all four flags. If any one is missing, it isn’t binomial.
1. Only two outcomes: success or failure.
2. A fixed number of trials, $n$.
3. The same chance of success on every trial.
4. Trials don’t affect each other.
(A) Heads or not, 12 tosses, the same chance each toss, and tosses don’t affect each other.
(B) Left-handed or not, 100 people. In a large population, each person picked has about the same chance of being left-handed.
(C) Hit or miss, 20 shots, and the question says each shot is independent.
All three pass the four flags.
In (D) a drawn marble stays out, so the bag changes. The chance of red on the second draw depends on the first.
$\text{after a red: }\tfrac{4}{8}$$\text{after a blue: }\tfrac{5}{8}$
That breaks flags 3 and 4, so (D) isn’t binomial.
In (D), the marbles are drawn without replacement. The chance of red starts at $\tfrac{5}{9}$, then changes to $\tfrac{4}{8}$ or $\tfrac{5}{8}$ depending on the first draw.
The trials aren’t independent and the chance of success isn’t constant, so the number of red marbles isn’t binomial.
(A), (B) and (C) all have two outcomes, a fixed number of trials, the same chance each time and independent trials.
How Queensland students answered in 2022
From the QCAA Mathematical Methods subject report for 2022. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2022 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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