Question 3
A bag contains 10 buttons of the same shape and size in different colours: 5 blue, 3 green and 2 red.
If 3 buttons are randomly drawn from the bag, which probability can be calculated using the binomial distribution?
That’s it. Step through the hints below to see why. In 2023, 67.1% of Queensland students chose (A).
Not quite. Without replacement, each blue button drawn stays out, so the chance of blue changes: $\tfrac{5}{10}$, then $\tfrac{4}{9}$, then $\tfrac{3}{8}$. A binomial needs the same chance on every trial. In 2023, 19.4% of Queensland students chose (B).
Close. With replacement keeps the chance fixed, but this event needs three categories: green, red and blue. A binomial only has success and failure. If green is the success, red and blue are both failures, so it can’t pick out “1 red”. In 2023, 9.1% of Queensland students chose (C).
Not quite. This fails on two counts. Without replacement, the chances change after each draw. And the event needs three categories: red, blue and the green buttons that must not be drawn. In 2023, 3.8% of Queensland students chose (D).
Two outcomes, green or not green. Each button goes back, so the chance of green stays $\tfrac{3}{10}$ on every draw. All four flags hold, so this is $\text{Bin}\left(3,\tfrac{3}{10}\right)$.
Nothing has been drawn yet, so the chance of blue is $\tfrac{5}{10}$. Make a draw and watch the bag.
Each blue button drawn stays out, so the chance of blue changes from draw to draw. The chance isn’t the same on every trial, so this isn’t binomial.
The chance of green stays $\tfrac{3}{10}$, because each button goes back. But this event needs three categories: green, red and blue. A binomial only counts successes and failures, so it can’t tell “2 green and 1 red” from “2 green and 1 blue”.
Nothing has been drawn yet. The event already needs three categories, red, blue and green, so it isn’t two outcomes. Make a draw and watch the chance of red.
This fails on two counts. The drawn buttons stay out, so the chance of red changes, and the event needs three categories: red, blue and green.
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. Here that’s 3 draws.
3. The same chance on every trial.
4. Trials don’t affect each other.
Without replacement, a drawn button stays out. The bag changes, so the chance changes. In (B), the chance of blue goes from $\tfrac{5}{10}$ to $\tfrac{4}{9}$.
That breaks flags 3 and 4, which rules out (B) and (D).
(C) is with replacement, but “2 green and 1 red” needs three categories. If green is the success, red and blue are both failures, so a binomial can’t tell them apart.
(A) has success as green, two outcomes, three draws and a fixed $p=\tfrac{3}{10}$.
$$P(X=3)=\left(\tfrac{3}{10}\right)^3=\tfrac{27}{1000}$$
In (A), each draw is green or not green, and each button goes back, so the chance of green is $\tfrac{3}{10}$ on all three draws.
$$X\sim\text{Bin}\left(3,\tfrac{3}{10}\right)$$
$\displaystyle P(3\text{ green})=P(X=3)$$\displaystyle =\left(\tfrac{3}{10}\right)^3=\tfrac{27}{1000}$
(B) and (D) are without replacement, so the chance changes after each draw. (C) needs three categories, so it isn’t two outcomes.
How Queensland students answered in 2023
From the QCAA Mathematical Methods subject report for 2023. 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 2023 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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