Question 1
The probability of hitting a target in a particular binomial experiment is 0.72
Determine the mean of the number of hits if this experiment is repeated eight times.
Not quite. 1.61 is $np(1-p)=8\times0.72\times0.28\approx1.61$, the variance. The mean is $np$. In 2024, 1.9% of Queensland students chose (A).
Not quite. 2.24 is $8\times0.28$, the mean number of misses. The question asks about hits, and a hit has probability 0.72. In 2024, 2.2% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2024, 94.4% of Queensland students chose (C).
Not quite. 7.28 is $8-0.72$. The mean of a binomial multiplies the number of trials by the probability of success: $8\times0.72$. In 2024, 1.2% of Queensland students chose (D).
This is the question. The bars balance at $8\times0.72=5.76$ hits, option (C). The variance, about 1.61, is option (A). It measures the spread, not the centre.
With $p=0.28$ the mean is $8\times0.28=2.24$, option (B). In the question that is the mean number of misses, not hits.
The orange marker is the balance point of the bars, and it always sits at $np$. Slide $p$ up and the bars and the mean move right together.
Each trial is a hit or a miss, the chance of a hit stays at 0.72, and there are eight independent trials. So the number of hits is binomial.
$$X\sim\text{Bin}(8,\,0.72)$$
The mean of a binomial distribution is the number of trials times the probability of success.
$$E(X)=np$$
Don’t mix it up with the variance, $np(1-p)$.
$$E(X)=8\times0.72=5.76$$
It makes sense. A 72% hit rate over eight tries should give a bit under six hits on average.
$$X\sim\text{Bin}(8,\,0.72)$$
$$E(X)=np=8\times0.72=5.76$$
On average you expect 5.76 hits in eight trials.
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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