Question 4
Consider the Bernoulli distribution where the outcomes for rolling a six-sided die are a four and not rolling a four.
Determine the variance of the resulting Bernoulli distribution in this scenario.
Not quite. $0.02\dot{7}=\tfrac{1}{36}=p^2$, the square of the mean. The variance is $p(1-p)=\tfrac16\times\tfrac56$. In 2024, 6.1% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2024, 71.2% of Queensland students chose (B).
Not quite. $0.1\dot{6}=\tfrac16$ is $p$, the probability of rolling a four. That is the mean of the Bernoulli distribution, not its variance. In 2024, 9.8% of Queensland students chose (C).
Not quite. $0.8\dot{3}=\tfrac56$ is $1-p$, the probability of not rolling a four. The variance multiplies the two probabilities together. In 2024, 11.9% of Queensland students chose (D).
This is the die. Rolling a four is the success, so $p=\tfrac16$, and the variance is $\tfrac{5}{36}=0.13\dot{8}$, option (B).
Swap success and failure and $p=\tfrac56$, but $p(1-p)$ doesn’t change. The variance is still $\tfrac{5}{36}$.
The variance is largest at $p=0.5$, where the outcome is hardest to predict. It falls to 0 at both ends, where the outcome is certain.
A Bernoulli trial has two outcomes. Call rolling a four a success, $X=1$, and anything else a failure, $X=0$. One face out of six is a four.
$$p=P(X=1)=\tfrac16$$
The variance of a Bernoulli distribution is the probability of success times the probability of failure.
$$\text{Var}(X)=p(1-p)$$
It comes from $E(X^2)-[E(X)]^2=p-p^2$.
$\displaystyle \text{Var}(X)=\tfrac16\times\tfrac56$$\displaystyle =\tfrac{5}{36}$
As a decimal, $\tfrac{5}{36}=0.13888\ldots$, and the 8 repeats.
$$\tfrac{5}{36}=0.13\dot{8}$$
$$p=\tfrac16$$
$\displaystyle \text{Var}(X)=p(1-p)$$\displaystyle =\tfrac16\times\tfrac56$$\displaystyle =\tfrac{5}{36}$$\displaystyle =0.13\dot{8}$
How Queensland students answered in 2024
From the QCAA Mathematical Methods subject report for 2024. 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 2024 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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