Question 6
If the probability of success in a Bernoulli trial is 0.30, the variance is
Not quite. 0.70 is the probability of failure, $1-p=1-0.30$. It is one of the two numbers you multiply, not the variance itself: $p(1-p)=0.30\times0.70$.
Not quite. 0.46 is the standard deviation, $\sqrt{0.21}\approx0.458$, rounded. The question asks for the variance, which is the value before you take the square root.
Not quite. 0.30 is the mean. For a Bernoulli trial $E(X)=p$, but the variance is $p(1-p)$.
That’s it. Step through the hints below to see why.
The bars show one trial: failure on the left, success on the right. The curve is the variance for every value of $p$, and the dot is the trial you have chosen. The dashed lines are the four options.
This is the question’s trial, $p=0.30$. The variance is $0.30\times0.70=0.21$ exactly, so the dot sits on the (D) line. That’s option (D).
The variance is 0.21 here too, because $p(1-p)$ stays the same when you swap $p$ and $1-p$. But the question has $p=0.30$, so here the bars are the wrong way round.
The dot looks close to the (D) line, but it isn’t on it. The variance here is , not 0.21. Use the button below.
This is the largest a Bernoulli variance can be, $0.5\times0.5=0.25$. The curve never reaches the (A), (B) or (C) lines.
The variance here is , the height of the dot. Wherever you move it, the dot stays at or below 0.25.
A Bernoulli trial is one trial with two outcomes. Success has probability $p$ and failure has probability $1-p$.
$p=0.30$$1-p=0.70$
The variance of a Bernoulli trial is the product of the two probabilities.
$\operatorname{Var}(X)=p(1-p)$$=0.30\times0.70$$=0.21$
Without a calculator, think $3\times7=21$, so $0.3\times0.7=0.21$.
Each wrong option is another number from the same trial.
(A) 0.70 is $1-p$. (C) 0.30 is the mean, $E(X)=p$. (B) 0.46 is the standard deviation, $\sqrt{0.21}\approx0.46$.
Sense check: $p(1-p)$ is never more than 0.25, so a Bernoulli variance can’t be 0.30, 0.46 or 0.70.
$\operatorname{Var}(X)=p(1-p)$$=0.30\times0.70$$=0.21$
Multiply the probability of success by the probability of failure. Don’t take the square root: that gives the standard deviation.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2020 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2020, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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