Question

Two loaded dice, each have the property that $2$ or $4$ is three times as likely to appear as $1,3,5$ or $6$ on each roll. When two such dice are rolled, the probability of obtaining a total of $7$ is $k$. Then the value of $[\frac{1}{k}]$ is , where $\left[x\right]$ represents the greatest integer less than or equal to $x$.

Answer 1

Let the probability of the outcome of $2$ is $p$
$1=p+p+\frac{p}{3}+\frac{p}{3}+\frac{p}{3}+\frac{p}{3}\Rightarrow p=\frac{3}{10}$
Then the probability of obtaining a total of $7$ is
$2(p\times \frac{p}{3}+p\times \frac{p}{3}+\frac{p}{3}\times \frac{p}{3})=\frac{14p^2}{9}=\frac{7}{50}\therefore k=\frac{7}{50}$