====== Question 5 and 6 Exercise 6.5 ======
Solutions of Question 5 and 6 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

=====Question 5=====
A student finds that probability of passing an algebra rest is $\dfrac{8}{9}$. What is the probability of failing the test?
====Solution====
Here we can differentiate as:
Let $$E=\{ event\, passing\, the\, test \}$$
and $$E^{\prime}=\{ event\, failing\, the\, test \}$$
$E$ and $E^{\prime}$ are complementary events, and we are given $P(E)=\dfrac{8}{9}$, therefore, \begin{align}P(E^{\prime})&=1-P(E)=1-\dfrac{8}{9}=\dfrac{1}{9}\end{align}

=====Question 6=====
In the two dice experiment, given that the the first dice shows $4$, what is the probability that the second dice shows a number greater than $4$?
====Solution====
The sample space of throwing the second dice is
\begin{align}S&=\{1,2,3,4,5,6\}\\
n(S)&=6 \end{align}
Let \begin{align}A&=\{ a\, number\, greater\, tha, 4\}\\
&=\{5,6\}\\
\text{then} n(A)&=2\end{align}
Thus, the probability that the second dice shows a number greater than $4$ is:
\begin{align}P(A)&=\dfrac{n(A)}{n(S)}\\
&=\dfrac{2}{6}\\
&=\dfrac{1}{3}\end{align}




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