Question

An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number $2$ times is equal to the probability of getting an even number $3$ times, then the probability of getting an odd number for odd number of times is :

• A. $\frac{3}{16}$
• B. $\frac{1}{2}$
• C. $\frac{5}{16}$
• D. $\frac{1}{32}$

Answer 1

The correct option is B $\frac{1}{2}$

$P$ (odd number twice) $=P$ (even number thrice)
$\Rightarrow {}^n{C}_2{\left(\frac{1}{2}\right)}^n={}^n{C}_3{\left(\frac{1}{2}\right)}^n\Rightarrow n=5$

Success is getting an odd number odd times
Hence $P$ (odd successes) $=P\left(1\right)+P\left(3\right)+P\left(5\right)$
$={}^5{C}_1{\left(\frac{1}{2}\right)}^5+{}^5{C}_3{\left(\frac{1}{2}\right)}^5+{}^5{C}_5{\left(\frac{1}{2}\right)}^5$
$=\frac{16}{2^5}=\frac{1}{2}$