Question
[Maximum mark: 5]
Alex works as a health insurance agent for Medical Benefits Fund. The probability that he succeeds in selling an insurance policy to a given customer aged 25 or older is $0.45$.
On a given day, he interacts with 8 customers in this age range. Find the probability that he will sell
(a) at least one insurance policy on this day; [3]
(b) exactly 2 insurance policies on this day. [2]
Answer/Explanation
(a) Let $X$ be the number of insurance policies that Alex sells on this day. Then we have $X \sim \mathrm{B}(8,0.45)$. This gives
$
\begin{aligned}
\operatorname{Pr}(X \geq 1) & =1-\operatorname{Pr}(X=0) \\
& =1-\operatorname{BinomPdf}(8,0.45,0) \\
& \approx 0.992 \quad \text { [by using G.D.C.] }
\end{aligned}
$
(b) The probability that he will sell exactly 2 insurance policies on the day is
$
\begin{aligned}
\operatorname{Pr}(X=2) & =\operatorname{BinomPdf}(8,0.45,2) \\
& \approx 0.157 \quad[\text { by using G.D.C.] }
\end{aligned}
$
Question
[Maximum mark: 6]
Peter has two water tanks with goldfish inside. The first tank is in the shape of a cylinder with diameter $40 \mathrm{~cm}$ and height $45 \mathrm{~cm}$. The second tank is in the shape of a cuboid with length $40 \mathrm{~cm}$, width $32 \mathrm{~cm}$, and height $42 \mathrm{~cm}$.
(a) Calculate the volume, giving your answer in $\mathrm{cm}^3$ correct to three significant figures,
(i) of the first water tank;
(ii) of the second water tank. [4]
Each goldfish requires $15000 \mathrm{~cm}^3$ of fresh water for a comfortable life.
(b) Calculate the number of goldfish Peter can safely put into his tanks. [2]
Answer/Explanation
(a) (i) Using the volume formula for a cylinder, we get
$
\begin{aligned}
V_{\text {cylinder }} & =\pi r^2 h \\
& =\pi\left(20^2\right)(45) \\
& \approx 56500 \mathrm{~cm}^3
\end{aligned}
$
(ii) Using the volume formula for a cuboid, we get
$
\begin{aligned}
V_{\text {cuboid }} & =L w h \\
& =40(32)(42) \\
& =53760 \\
& \approx 53800 \mathrm{~cm}^3
\end{aligned}
$
(b) To find the number of goldfish in the first tank we divide $V_1$ by 15000 and get
$
\begin{aligned}
\frac{V_1}{15000} & =\frac{56549}{15000} \\
& =3.77 \\
& =3 \text { goldfish. }
\end{aligned}
$