Question

If two loaded dice each have the property that $2$ and $4$ is three times as likely to appears as $1,3,5\text{or}6$ on each roll. When two such dice are rolled, the probability of obtaining a total of $7$ is $p$, then the value of $[\frac{1}{p}]$ is (where $\left[x\right]$ represents the greates integer less than or equal to $x$.)

Answer 1

Let the probability of the faces $1,3,5\text{or}6$ be $p$ for each face.
Hence, probability of each of the faces $2$ or $4$ is $3p$
Now according to the question $4p+6p=1$
$\Rightarrow p=\frac{1}{10}$
$\therefore P\left(1\right)=P\left(3\right)=P\left(5\right)=P\left(6\right)=\frac{1}{10}$
and $P\left(2\right)=P\left(4\right)=\frac{3}{10}$
$\Rightarrow$ recquired probability
$p=P\left(\text{total of 7 with a draw of dice}\right)=P(16,61,25,52,43,34)=2(\frac{1}{10}\cdot \frac{1}{10})+2(\frac{3}{10}\cdot \frac{1}{10})+2(\frac{3}{10}\cdot \frac{1}{1})=\frac{2+6+6}{100}=\frac{14}{100}=\frac{7}{50}$