Question

If 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 $p$, then the value of $\left[\frac{1}{p}\right]$ is, where $[\cdot ]$ represents the greatest integer function

Answer 1

Let the probability of the faces $1,3,5$ or $6$ be $k$ for each face
Hence, probability of each of the faces $2$ or $4$ is $3k$
Now, according to the question $4k+6k=1$
$\Rightarrow k=\frac{1}{10}$
$\therefore P\left(1\right)=P\left(3\right)=P\left(5\right)=P\left(6\right)=\frac{1}{10}\text{and}P\left(2\right)=P\left(4\right)=\frac{3}{10}$
We have,
$p=P\left(\text{getting a total of 7}\right)$
$=P\left(\right(1,6),(6,1),(2,5),(5,2),(4,3),(3,4\left)\right)$
$=2(\frac{1}{10}.\frac{1}{10})+2(\frac{3}{10}.\frac{1}{10})+2(\frac{3}{10}.\frac{1}{10})$
$=\frac{2+6+6}{100}=\frac{14}{100}=\frac{7}{50}$
$\therefore \left[\frac{1}{p}\right]=\left[\frac{50}{7}\right]=7$