For a loaded die, it is given that

$$\begin{aligned} P(1) & =P(2)=0.2 \\ P(3)=P(5) & =P(6)=0.1 \text { and } P(4)=0.3 \end{aligned}$$

Also, die is thrown two times.

Here, $A=$ Same number each time and $B=$ Total score is 10 or more

$$\begin{aligned} &\begin{array}{lrl} \therefore & A & =\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)\} \\ \text { So, } & P(A) & =[P(1,1)+P(2,2)+P(3,3)+P(4,4)+P(5,5)+P(6,6)] \\ & & =[P(1) \cdot P(1)+P(2) \cdot P(2)+P(3) \cdot P(3)+P(4) \cdot P(4)+P(5) \cdot P(5)+P(6) \cdot P(6)] \\ & & =[0.2 \times 0.2+0.2 \times 0.2+0.1 \times 0.1+0.3 \times 0.3+0.1 \times 0.1+0.1 \times 0.1] \\ & & =0.04+0.04+0.01+0.09+0.01+0.01=0.20 \\ \text { and } & B & =\{(4,6),(6,4),(5,5),(5,6),(6,5),(6,6)\} \\ \therefore & P(B) & =P(4,6)+P(6,4)+P(5,5)+P(5,6)+P(6,5)+P(6,6) \\ & & =P(4) \cdot P(6)+P(6) \cdot P(4)+P(5) \cdot P(5)+P(5) \cdot P(6)+P(6) \cdot P(5)+P(6) \cdot P(6) \\ & & =0.3 \times 0.1+0.1 \times 0.3+0.1 \times 0.1+0.1 \times 0.1+0.1 \times 0.1+0.1 \times 0.1 \\ & & =0.03+0.03+0.01+0.01+0.01+0.01=0.10 \end{array}\\ &\text { Also, }\\ &\begin{aligned} & A \cap B=\{(5,5),(6,6)\} \\ & \therefore \quad P(A \cap B)=P(5,5)+P(6,6)=P(5) \cdot P(5)+P(6) \cdot P(6) \\ & =0.1 \times 0.1+0.1 \times 0.1=0.01+0.01=0.02 \end{aligned} \end{aligned}$$

We know that, for two events $A$ and $B$, if $P(A \cap B)=P(A) \cdot P(B)$, then both are independent events.

Here, $P(A \cap B)=0.02$ and $P(A) \cdot P(B)=0.20 \times 0.10=0.02$

Thus, $P(A \cap B)=P(A) \cdot P(B)=0.02$

Hence, $A$ and $B$ are independent events.

$$\begin{aligned} &\text { Referring to the above solution, we have }\\ &\begin{array}{rlrl} & A =\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)\} \\ \Rightarrow & n(A) =6 \text { and } n(S)=6^2=36 & \text { [where, } S \text { is sample space] } \\ \therefore & P(A) =\frac{n(A)}{n(S)}=\frac{6}{36}=\frac{1}{6} \end{array} \end{aligned}$$

$$\begin{array}{lrl} \text { and } & B & =\{(4,6),(6,4),(5,5),(6,5),(5,6),(6,6)\} \\ \Rightarrow & n(B) & =6 \text { and } n(S)=6^2=36 \\ \therefore & P(B) & =\frac{n(B)}{n(S)}=\frac{6}{36}=\frac{1}{6} \end{array}$$

$$\begin{array}{ll} \text { Also, } & A \cap B=\{(5,5),(6,6)\} \\ \Rightarrow & n(A \cap B)=2 \text { and } n(S)=36 \\ \therefore & P(A \cap B)=\frac{2}{36}=\frac{1}{18} \\ \text { Also, } & P(A) \cdot P(B)=\frac{1}{36} \end{array}$$

Thus, $\quad P(A \cap B) \neq P(A) \cdot P(B) \quad\left[\because \frac{1}{18} \neq \frac{1}{36}\right]$

So, we can say that both $A$ and $B$ are not independent events.

$$\begin{array}{ll} \text { Thus, } & P(A \cup B)=0.6 \text { and } P(A \cap B)=0.3 \\ \text { Also, } & P(A \cup B)=P(A)+P(B)-P(A \cap B) \\ \Rightarrow & 0.6=P(A)+P(B)-0.3 \end{array}$$

$$\begin{array}{lrr} \Rightarrow & P(A)+P(B)=0.9 & \\ \Rightarrow & {[1-P(\bar{A})]+[1-P(\bar{B})]=0.9} & {[\because P(A)=1-P(\bar{A}) \text { and } P(B)=1-P(\bar{B})]} \\ \Rightarrow & P(\bar{A})+P(\bar{B})=2-0.9=1.1 & \end{array}$$

Let $R=\{5$ red marbles $\}$ and $B=\{3$ black marbles $\}$

For atleast one of the three marbles drawn be black, if the first marble is red, then the following three conditions will be followed

(i) Second ball is black and third is red $\left(E_1\right)$.

(ii) Second ball is black and third is also black ( $E_2$ ).

(iii) Second ball is red and third is black $\left(E_3\right)$.

$$\begin{aligned} \therefore & P\left(E_1\right)=P\left(R_1\right) \cdot P\left(B_1 / R_1\right) \cdot P\left(R_2 / R_1 B_1\right)=\frac{5}{8} \cdot \frac{3}{7} \cdot \frac{4}{6}=\frac{60}{336}=\frac{5}{28} \\ & P\left(E_2\right)=P\left(R_1\right) \cdot P\left(B_1 / R_1\right) \cdot P\left(B_2 / R_1 B_1\right)=\frac{5}{8} \cdot \frac{3}{7} \cdot \frac{2}{6}=\frac{30}{336}=\frac{5}{56} \end{aligned}$$

$$ \begin{array}{ll} \text { and } & P\left(E_3\right)=P\left(R_1\right) \cdot P\left(R_2 / R_1\right) \cdot P\left(B_1 / R_1 R_2\right)=\frac{5}{8} \cdot \frac{4}{7} \cdot \frac{3}{6}=\frac{60}{336}=\frac{5}{28} \\ \therefore & P(E)=P\left(E_1\right)+P\left(E_2\right)+P\left(E_3\right)=\frac{5}{28}+\frac{5}{56}+\frac{5}{28} \end{array}$$

$=\frac{10+5+10}{56}=\frac{25}{56}$

$$\begin{aligned} &\text { Two dice are thrown together i.e., sample space }(S)=36 \Rightarrow n(S)=36\\ &E=\text { A total of } 4=\{(2,2),(3,1),(1,3)\} \end{aligned}$$

$$\begin{array}{l} \Rightarrow & n(E) =3 \\ & F =\text { A total of } 9 \text { or more } \\ & =\{(3,6),(6,3),(4,5),(4,6),(5,4),(6,4),(5,5),(5,6),(6,5),(6,6)\} \\ \Rightarrow & n(F) =10 \\ \end{array}$$

$G=$ a total divisible by $5=\{(1,4),(4,1),(2,3),(3,2),(4,6),(6,4),(5,5)\}$

$\Rightarrow \quad n(G)=7$

$$\begin{array}{lc} \text { Here, } & (E \cap F)=\phi \text { and }(E \cap G)=\phi \\ \text { Also, } & (F \cap G)=\{(4,6),(6,4),(5,5)\} \\ \Rightarrow & n(F \cap G)=3 \text { and }(E \cap F \cap G)=\phi \\ \therefore & P(E)=\frac{n(E)}{n(S)}=\frac{3}{36}=\frac{1}{12} \end{array}$$

$$\begin{aligned} P(F) & =\frac{n(F)}{n(S)}=\frac{10}{36}=\frac{5}{18} \\ P(G) & =\frac{n(G)}{n(S)}=\frac{7}{36} \\ P(F \cap G) & =\frac{3}{36}=\frac{1}{12} \\ \text{and}\quad P(F) \cdot P(G) & =\frac{5}{18} \cdot \frac{7}{36}=\frac{35}{648} \end{aligned}$$

Here, we see that $P(F \cap G) \neq P(F) \cdot P(G)$

[since, only $F$ and $G$ have common events, so only $F$ and $G$ are used here]

Hence, there is no pair which is independent.