Let B_i (i = 1, 2, 3) be three independent events in a sample space. The probability that only B₁ occur is $\alpha$, only B₂ occurs is $\beta$ and only B₃ occurs is $\gamma$. Let p be the probability that none of the events B_i occurs and these 4 probabilities satisfy the equations $\left( {\alpha - 2\beta } \right)p = \alpha \beta$ and $\left( {\beta - 3\gamma } \right)p = 2\beta \gamma$ (All the probabilities are assumed to lie in the interval (0, 1)).
Then ${{P\left( {{B_1}} \right)} \over {P\left( {{B_3}} \right)}}$ is equal to ________.

Let x, y, z be probability of B₁, B₂, B₃ respectively <br><br>$\alpha$ = P(B₁ $\cap$ $\overline {{B_2}} \cap \overline {{B_3}}$) = $$P\left( {{B_1}} \right)P\left( {\overline {{B_2}} } \right)P\left( {\overline {{B_3}} } \right)$$ <br><br>$\Rightarrow$ x(1 $-$ y)(1 $-$ z) = $\alpha$<br><br>Similarly, y(1 $-$ x)(1 $-$ z) = $\beta$<br><br> z(1 $-$ x)(1 $-$ y) = $\gamma$<br><br>and (1 $-$ x)(1 $-$ y)(1 $-$ z) = p<br><br>($\alpha$ $-$ 2$\beta$)p = $\alpha$$\beta$<br><br>(x(1 $-$ y)(1 $-$ z) $-$2y(1 $-$ x)(1 $-$ z)) (1 $-$ x)(1 $-$ y)(1 $-$ z) = xy(1 $-$ x)(1 $-$ y)(1 $-$ z)<br><br>x $-$ xy $-$ 2y + 2xy = xy<br><br>x = 2y ...... (1)<br><br>Similarly ($\beta$ $-$ 3$\gamma$)p = 2$\beta$$\gamma$<br><br>$\Rightarrow$ y = 3z .... (2)<br><br>From (1) &amp; (2)<br><br>x = 6z<br><br>Now <br><br>${x \over z} = 6$