Question

A die is loaded so that the probability of face $i$ is proportional to $i,i=1,2,3,...,6$. The probability of an even number occurring when the die is rolled, is

• A. $\frac{2}{7}$
• B. $\frac{4}{7}$
• C. $\frac{3}{7}$
• D. None of these

Answer 1

The correct option is B $\frac{4}{7}$

$P\left(1\right)=1\times k=k$; where $k$ is the proportionality constant.
$P\left(2\right)=2k$
$P\left(3\right)=3k$
$P\left(4\right)=4k$
$P\left(5\right)=5k$
$P\left(6\right)=6k$.
Hence Total $=21k$.
Now probability that the face which turns up has even number
$=\frac{P\left(2\right)+P\left(4\right)+P\left(6\right)}{P\left(total\right)}$
$=\frac{2k+4k+6k}{21k}$
$=\frac{12}{21}$
$=\frac{4}{7}$.
Also $P\left(total\right)$ has to be $1$.
Hence $21k=1$
$\therefore k=\frac{1}{21}$.