Question 6
When seeds of a certain variety of flower are planted, the probability of each seed germinating is 0.8.
If eight seeds are planted, what is the probability that at least six seeds will germinate?
That’s it. Step through the hints below to see why. In 2021, 61.4% of Queensland students chose (A).
Not quite. 0.503 is $P(X\ge7)$, seven or eight seeds. That is “more than six”. At least six includes exactly six as well, so add the bar for $X=6$. In 2021, 13.0% of Queensland students chose (B).
Not quite. 0.294 is $P(X=6)$, exactly six seeds. At least six also counts seven and eight, so add $P(X=7)$ and $P(X=8)$. In 2021, 24.2% of Queensland students chose (C).
Not quite. 0.001 is what you get with 0.2 as the probability of success. That is the chance that at least six seeds fail to germinate. Each seed germinates with probability 0.8. In 2021, 0.9% of Queensland students chose (D).
This is the question. At least six means 6, 7 or 8 seeds, so add those three bars. The total is 0.797.
This is $P(X\ge7)=0.503$, option (B). It is “more than six”. It leaves out the bar for exactly six, which at least six includes.
This is only the bar for exactly six, 0.294, option (C). At least six also includes seven and eight.
Here the probability of success is 0.2, the chance a seed fails. So this is the chance that at least six seeds fail, about 0.001, option (D). Germinating has probability 0.8.
At least shades bar and every bar to its right. Those bars add to .
At least 8 means all eight seeds, so only bar 8 is shaded: .
Exactly is a single bar, . The nine bars always add to 1.
Each of the eight seeds either germinates or doesn’t, with the same probability 0.8 each time. Let $X$ be the number that germinate.
$$X\sim\text{Bin}(8,\,0.8)$$
At least six means six or more. Out of eight seeds, that is 6, 7 or 8.
$\displaystyle P(X\ge6)$$\displaystyle =P(X=6)$$\displaystyle +P(X=7)$$\displaystyle +P(X=8)$
Work out each bar with $\binom{8}{x}(0.8)^x(0.2)^{8-x}$ and add.
$\displaystyle P(X\ge6)$$\displaystyle \approx0.2936$$\displaystyle +0.3355$$\displaystyle +0.1678$$\displaystyle \approx0.797$
On your calculator, use the binomial cumulative (cdf) function with $n=8$, $p=0.8$, lower bound 6 and upper bound 8.
$$X\sim\text{Bin}(8,\,0.8)$$
$\displaystyle P(X\ge6)$$\displaystyle =P(X=6)$$\displaystyle +P(X=7)$$\displaystyle +P(X=8)$
$\displaystyle \approx0.29360$$\displaystyle +0.33554$$\displaystyle +0.16777$$\displaystyle \approx0.7969$$\displaystyle \approx0.797$
How Queensland students answered in 2021
From the QCAA Mathematical Methods subject report for 2021. 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 2021 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
Spot a mistake on this page? Report it