Let $B_i\left(i=1,2,3\right)$ be three independent events in a sample space. The probability that only $B_1$ occur is $\alpha ,$ only $B_2$ occurs is $\beta$ and only $B_3$ 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 $\left.\left(0,1\right)\right)$ Then $\frac{P\left(B_1\right)}{P\left(B_3\right)}$ is equal to______.

Let there be three independent events $E_1,E_2$ and $E_3.$ The probability that only $E_1$ occurs is $\alpha$ only $E_2$ occurs is $\beta$ and only $E_3$ occurs is $\gamma .$ Let ${}^{\text{'}}p^{\text{'}}$ denote the probability of none of events occurs that satisfies the equations $(\alpha -2\beta )\mathrm{p}=\alpha \beta$ and $(\beta -3\gamma )\mathrm{p}=2\beta \gamma .$ All the
given probabilities are assumed to lie in the interval $\left(0, 1\right).$

Then, $\frac{\text{ Probability of occurrence of }{E}_1}{\text{ Probability of occurrence of }{E}_3}$ is equal to ________.