Question

An unbiased die is tossed until a number greater than 4 appears. The probability that an even number of tosses is needed is

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

Answer 1

Answer: (B)

$\frac{2}{5}$

Let $E$ denote the event that a number greater than $4$ appears, then
$P\left(E\right)=\frac{2}{6}=\frac{1}{3}$ and $P\left(E^{\prime }\right)=\frac{2}{3}$
$P$( even number of tosses are required)
$=P$(E'E$or$E'E'E'E'$or$E'E'E'E'E'E'$or$...)
$P\left(E^{\prime }\right)P\left(E\right)+P{\left(E^{\prime }\right)}^{3}P\left(E\right)+P{\left(E^{\prime }\right)}^{s}P\left(E\right)+...$
$=\frac{P\left(E^{\prime }\right)P\left(E\right)}{1-P{\left(E^{\prime }\right)}^2}=\frac{\frac{2}{9}}{1-\frac{4}{9}}=\frac{2}{5}$