Consider an experiment of tossing a coin repeatedly until the...

Given,

$P ( H ) = \dfrac{1}{3}$

So, $P ( T ) = 1 - \dfrac{1}{3} = \dfrac{2}{3}$

Now probability of the event is given by,

$P ( E ) = P ( HH ) + P ( THH ) + P ( HTHH ) + P ( THTHH ) + P ( HTHTHH ) + P ( THTHTHH ) + \ldots$

$⇒ P ( E ) = \dfrac{1}{3} \times \dfrac{1}{3} + \dfrac{2}{3} \times \dfrac{1}{3} \times \dfrac{1}{3} + \dfrac{1}{3} \times \dfrac{2}{3} \times \dfrac{1}{3} \times \dfrac{1}{3} + \dfrac{4}{{3}^{5}} + \dfrac{4}{{3}^{6}} + \dfrac{8}{{3}^{7}} + \dfrac{8}{{3}^{8}} + \ldots$

$⇒ P ( E ) = \dfrac{1}{{3}^{2}} + \dfrac{2}{{3}^{3}} + \dfrac{2}{{3}^{4}} + \dfrac{4}{{3}^{5}} + \dfrac{4}{{3}^{6}} + \dfrac{8}{{3}^{7}} + \dfrac{8}{{3}^{8}} + \ldots$

$⇒ P ( E ) = \left(\dfrac{1}{{3}^{2}} + \dfrac{2}{{3}^{4}} + \dfrac{4}{{3}^{6}} + \ldots .\right) + \left(\dfrac{2}{{3}^{3}} + \dfrac{4}{{3}^{5}} + \dfrac{8}{{3}^{7}} + \ldots .\right)$

$⇒ P ( E ) = \left(\dfrac{\dfrac{1}{{3}^{2}}}{\left(1 - \dfrac{2}{{3}^{2}}\right)}\right) + \left(\dfrac{\dfrac{2}{{3}^{3}}}{\left(1 - \dfrac{2}{{3}^{2}}\right)}\right)$

$⇒ P ( E ) = \dfrac{1}{7} + \dfrac{2}{21} = \dfrac{5}{21}$