# Two dice are thrown. The events A, B and C are as follows:

Question:

Two dice are thrown. The events A, B and C are as follows:

A: getting an even number on the first die.

B: getting an odd number on the first die.

C: getting the sum of the numbers on the dice ≤ 5

(i) A and B are mutually exclusive

(ii) A and B are mutually exclusive and exhaustive

(iii) $\mathrm{A}=\mathrm{B}^{\prime}$

(iv) $\mathrm{A}$ and $\mathrm{C}$ are mutually exclusive

(v) $A$ and $B^{\prime}$ are mutually exclusive

(vi) $\mathrm{A}^{\prime}, \mathrm{B}^{\prime}, \mathrm{C}$ are mutually exclusive and exhaustive.

Solution:

$A=\left\{\begin{array}{l}(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3), \\ (4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}$

$\mathrm{B}=\left\{\begin{array}{l}(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(3,1),(3,2),(3,3), \\ (3,4),(3,5),(3,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)\end{array}\right\}$

$\mathrm{C}=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1)\}$

(i) It is observed that $A \cap B=\Phi$

$\therefore \mathrm{A}$ and $\mathrm{B}$ are mutually exclusive.

Thus, the given statement is true.

(ii) It is observed that $A \cap B=\Phi$ and $A \cup B=S$

$\therefore \mathrm{A}$ and $\mathrm{B}$ are mutually exclusive and exhaustive.

Thus, the given statement is true.

(iii) It is observed that

$\mathrm{B}^{\prime}=\left\{\begin{array}{l}(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3), \\ (4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}=\mathrm{A}$

Thus, the given statement is true.

(iv) It is observed that $A \cap C=\{(2,1),(2,2),(2,3),(4,1)\} \neq \varnothing$

$\therefore$ A and $C$ are not mutually exclusive.

Thus, the given statement is false.

(v) $\mathrm{A} \cap \mathrm{B}^{\prime}=\mathrm{A} \cap \mathrm{A}=\mathrm{A}$

$\therefore \mathrm{A} \cap \mathrm{B}^{\prime} \neq \phi$

$\therefore \mathrm{A}$ and $\mathrm{B}^{\prime}$ are not mutually exclusive.

Thus, the given statement is false.

(vi) It is observed that $\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime} \cup \mathrm{C}=\mathrm{S}$;

However, $\mathrm{B}^{\prime} \cap \mathrm{C}=\{(2,1),(2,2),(2,3),(4,1)\} \neq \phi$

Therefore, events $A^{\prime}, B^{\prime}$ and $C$ are not mutually exclusive and exhaustive.

Thus, the given statement is false.