જો A અને B બે ઘટનાઓ છે કે જેથી Pleft( { | vedclass

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Question

Let $A$ and $B$ be two events such that

$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

and $\mathrm{P}(\mathrm{

Vedclass · Accepted Answer

$P\left( A \right) + P\left( B \right) = 1$

Vedclass · Answer

Let $A$ and $B$ be two events such that

$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

and $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

option (a) : since $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

(given) 

therefore $A$ and $B$ are equally likely

Suppose option $(b)$ and option $(c)$ are correct.

$	herefore \mathrm{P}\left(\mathrm{A} \cap \mathrm{B}^{\prime}ight)=0$ and $\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}ight)=0$

$\Rightarrow \mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$

and $P(B)-P(A \cap B)=0$

$\Rightarrow P(A)=P(A \cap B)$

and $\mathrm{P}(\mathrm{B})=\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

Thus $P(A)=P(B)=P(A \cap B)$

$=\mathrm{P}(\mathrm{A} \cup \mathrm{B})$

$[\because 	ext { Given } \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A} \cup \mathrm{B})]$

Also, we know

$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

$=\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{A} \cap \mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

$=\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

which is true from given condition 

Hence, option $(a),(b)$ and $(c)$ are correct.

જો $A$ અને $B$ બે ઘટનાઓ છે કે જેથી $P\left( {A \cup B} \right) = P\left( {A \cap B} \right)$, તો આપેલ પૈકી કયું વિધાન અસત્ય છે .

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