Long Questions (Question 9)

Question 9:
(a) 30% of the pens in a box are blue. Charlie picks 4 pens at random. Find the probability that at least one pen picked is not blue.

(b) The mass of papayas harvested from an orchard farm follows a normal distribution with a mean of 2 kg and a standard deviation of h kg. It is given that 15.87% of the papayas have a mass more than 2.5 kg.
(i) Calculate the value of h.
(ii) Given the number of papayas harvested from the orchard farm is 1320, find the number of papayas that have the mass between 1.0 kg and 2.5 kg.

Solution:
$\begin{array}{l}\text{(a)}\\ P\left(X\ge 1\right)=1-P\left(X=0\right)\\ =1-{}^{4}C{}_4{\left(0.3\right)}^4{\left(0.7\right)}^0\\ =0.9919\end{array}$

$\begin{array}{l}\text{(b) }\mu =2,\sigma =h\\ \left(i\right)\\ P\left(X>2.5\right)=15.87\%\\ P\left(Z>\frac{2.5-2}{h}\right)=0.1587\\ P\left(Z>1.0\right)=0.1587\\ \frac{2.5-2}{h}=1.0\\ h=0.5\end{array}$


$\begin{array}{l}\left(ii\right)\\ p=P\left(1.0<x<2.5\right)\\ =P\left(\frac{1.0-2}{0.5}<Z<\frac{2.5-2}{0.5}\right)\\ =P\left(-2<Z<1\right)\\ =1-P\left(Z<-2\right)-P\left(Z>1\right)\\ =1-P\left(Z>2\right)-P\left(Z>1\right)\\ =1-0.0228-0.1587\\ =0.8185\\ \\ \text{Number of papayas}=0.8185\times 1320\\ =1080\end{array}$