Question

A die is loaded so that the probability of face i is proportional to $i$, $i=1,2,...,6$. The probability of an even number occuring when die is rolled is $\frac{k}{7}$. Find the value of $k$.

Answer 1

Since the probability of the faces are proportional to the numbers on them,
we can take the probabilities of faces $1,2,6$ as $k,2k,......,6k$ respectively.
Since on of the faces must occur, we have
$k+2k+3k+4k+5k+6k+=1$
$\Rightarrow k=\frac{1}{21}$
$\therefore$The probability of an even number $=2k+4k+6k=$ $12k=12\times \frac{1}{21}=\frac{4}{7}$