Long Questions (Question 8)

Question 8:
(a) A survey is carried out about red crescents in a school.
It is found that the mean of the number of red crescents is 315, the variance is 126 and the probability that a student participate in red crescents is p.
(i) Find the value of p.
(ii) If 8 students from the school are chosen at random, find the probability that more than 5 students participate in red crescents.

(b) The mass of fertiliser used in an orchard farm is normally distributed with mean 5 kg and variance 0.8 kg. Find the probability that on a particular day, more than 6 kg of fertiliser is used.

Solution:
$\begin{array}{l}\text{(a)(i)}\\ np=315\\ np\left(1-p\right)=126\\ 315\left(1-p\right)=126\\ 1-p=\frac{126}{315}\\ 1-p=0.4\\ p=0.6\\ \\ \left(ii\right)\\ n=8,p=0.6\\ P(X=r)={}^{n}c{}_r.p^r.q^{n-r}\\ P(X=r)={}^{8}C{}_r{\left(0.6\right)}^r{\left(0.4\right)}^8\\ \\ P(X>5)\\ =P\left(X=6\right)+P\left(X=7\right)+P\left(X=8\right)\\ ={}^{8}C{}_6{\left(0.6\right)}^6{\left(0.4\right)}^2+{}^{8}C{}_7{\left(0.6\right)}^7{\left(0.4\right)}^1+{}^{8}C{}_8{\left(0.6\right)}^8{\left(0.4\right)}^0\\ =0.20902+0.08958+0.01680\\ =0.3154\end{array}$

$\begin{array}{l}\text{(b)}\\ P\left(X>6\right)=P\left(Z>\frac{6-5}{\sqrt{0.8}}\right)\\ =P\left(Z>1.12\right)\\ =0.1314\end{array}$