====== Question 3 and 4 Exercise 6.5 ======
Solutions of Question 3 and 4 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 3=====
Given $P(A)=0.5$ and $P(A \cup B)=0.6$. find $P(B)$ if $A$ and $B$ are mutually exclusive.
====Solution====
We are given that $\mathrm{A}$ and $B$ are mutually exclusive, therefore $A \cap B=\emptyset$.

Thus \begin{align}P(A \cup B)&=P(A)+P(B)\\
\Rightarrow P(B)&=P(A \cup B)-P(A)\\
&=0.6-.0 .5=0.1 \end{align}

=====Question 4=====
A bag contains $30$ tickets numbered from $1$ to $30.$ One ticket is selected at random. Find the probability that its number is either odd or the square of an integer.
====Solution====
Total numbers written on tickets are
\begin{align}S&=\{1,2,3, \ldots, 50\} \text { so }\\
n(S)&=50 \end{align}
Let \begin{align}A \{odd \,numbers \}&=\{1,3,5,..,29\}\\
n(A)&=15\\
\text{Let}\,
B&=\{ number \,square \,of\, an\, integer \}\\
&=\{1,4,9,16,25\}\\
n(B)&=5\end{align}
Álso \begin{align}A \cap B&=\{1,9,25\}\\
\text{Therefore} n(A \cap B)&=3\end{align}
Now $$P(A)=\dfrac{15}{30}, P(B)=\dfrac{5}{30}, P(A \cap B)=\dfrac{3}{30}$$
The probability that the number is either odd or the square of an integer is:
\begin{align}P(A \cup B)&=P(A)+P(B)-P(A \cap B) \\
& =\dfrac{15}{30}+\dfrac{5}{30}-\dfrac{3}{30}=\dfrac{17}{30}\end{align}







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