7.5.1 Probability, Paper 1 (Short Questions)

$\begin{array}{l}\text{Probability (both of the students are chosen as the school prefect})\\ =\frac{3}{4}\times \frac{5}{6}\\ =\frac{5}{8}\end{array}$

(b)
$\begin{array}{l}\text{Probability (only one student is chosen as a school prefect})\\ =\left(\frac{3}{4}\times \frac{1}{6}\right)+\left(\frac{1}{4}\times \frac{5}{6}\right)\\ =\frac{3}{24}+\frac{5}{24}\\ =\frac{1}{3}\end{array}$

Question 3:
A sample space of an experiment is given by S = {1, 2, 3, … , 21}. Events Q and R are defined as follows:
Q : {3, 6, 9, 12, 15, 18, 21}
R : {1, 3, 5, 15, 21}

Find
(a) P(Q)
(b) P(Q and R)

Solution:
(a)

$\begin{array}{l}n\left(S\right)=21,n\left(Q\right)=7\\ P\left(Q\right)=\frac{7}{21}=\frac{1}{3}\end{array}$

(b)
$\begin{array}{l}Q\cap R=\left\{3,15,21\right\},\\ thenn\left(Q\cap R\right)=3\\ P\left(Q\text{ and }R\right)=P\left(Q\cap R\right)\\ =\frac{n\left(Q\cap R\right)}{n\left(S\right)}\\ =\frac{3}{21}\\ =\frac{1}{7}\end{array}$

Question 4:
The events A and B are not independent.
$\begin{array}{l}\text{Given }P\left(A\right)=\frac{3}{5},P\left(B\right)=\frac{1}{4}\text{ and }P\left(A\cup B\right)=\frac{1}{5},\text{ find}\\ \text{(a) }P\left[\left(A\cup B\right)\text{'}\right],\\ \text{(b) }P\left(A\cap B\right).\end{array}$

Solution:
(a)
$\begin{array}{l}P\left[\left(A\cup B\right)\text{'}\right]=1-P\left(A\cup B\right)\\ =1-\frac{1}{5}\\ =\frac{4}{5}\end{array}$

(b)
$\begin{array}{l}P\left(A\cap B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cup B\right)\\ =\frac{3}{5}+\frac{1}{4}-\frac{1}{5}\\ =\frac{13}{20}\end{array}$