QJEEM21Ma0477: JEE Main 2021 Mathematics - Class 12 JEEM21Ma0477 Solution: Let there be three independent events E 1 , E 2 and E 3 . The…

Let P(E₁) = x, P(E₂) = y and P(E₃) = z

$\alpha$ = P $\left( {{E_1} \cap {{\overline E }_2} \cap {{\overline E }_3}} \right)$ = $P\left( {{E_1}} \right).P\left( {{{\overline E }_2}} \right).P\left( {{{\overline E }_3}} \right)$

$\Rightarrow$ $\alpha$ = x(1 $-$ y) (1 $-$ z) ......(i)

Similarly

β = (1 – x).y(1 – z) ...(ii)

$\gamma$ = (1 – x)(1 – y).z ...(iii)

p = (1 – x)(1 – y)(1 – z) ...(iv)

From (i) and (iv)

${x \over {1 - x}} = {\alpha \over p}$

$\Rightarrow$ x = ${\alpha \over {\alpha + p}}$

From (iii) and (iv)

${z \over {1 - z}} = {\gamma \over p}$

$\Rightarrow$ z = ${\gamma \over {\gamma + p}}$

$\therefore$ ${{P\left( {{E_1}} \right)} \over {P\left( {{E_3}} \right)}} = {x \over z} = {{{\alpha \over {\alpha + p}}} \over {{\gamma \over {\gamma + p}}}}$ $= {{{{\gamma + p} \over \gamma }} \over {{{\alpha + p} \over \alpha }}} = {{1 + {p \over \gamma }} \over {1 + {p \over \alpha }}}$ ..(v)

Also given,

( $\alpha$ $-$ 2 $\beta$ )p = $\alpha$ $\beta$ $\Rightarrow$ $\alpha$ p = ( $\alpha$ + 2p) $\beta$ ....(vi)

$\beta$ $-$ 3 $\gamma$ )p = 2 $\beta$ $\gamma$ $\Rightarrow$ 3 $\gamma$ p = (p - 2 $\gamma$ ) $\beta$ .....(vii)

From (vi) and (vii),

${\alpha \over {3\gamma }} = {{\alpha + 2p} \over {p - 2\gamma }}$

$\Rightarrow$ p $\alpha$ - 6p $\gamma$ = 5 $\gamma$ $\alpha$

$\Rightarrow$ ${p \over \gamma } - {{6p} \over \alpha } = 5$

$\Rightarrow$ ${p \over \gamma } + 1 = 6\left( {{p \over \alpha } + 1} \right)$ ....(viii)

Now from (v) and (viii),

${{P\left( {{E_1}} \right)} \over {P\left( {{E_3}} \right)}}$ = 6