Question
Probability that India will win against Pakistan in a cricket match is $$\frac{2}{3},$$ in series of $$5$$ matches what is the probability that India will win the series ?
A.
$$\frac{{161}}{{81}}$$
B.
$$\frac{{192}}{{243}}$$
C.
$$\frac{{172}}{{243}}$$
D.
none of these
Answer :
$$\frac{{192}}{{243}}$$
Solution :
Total number of matches $$= n = 5$$
India will win the series if it wins either $$3$$ or $$4$$ or $$5$$ matches.
In previous question we have calculated the value of $$P\left( 3 \right) = $$ probability of winning $$3$$ matches
$$ = \left( {{}^5{C_3}} \right){\left( {\frac{2}{3}} \right)^3}{\left( {\frac{1}{3}} \right)^2}$$
Required Probability
$$\eqalign{
& = P\left( 3 \right) + P\left( 4 \right) + P\left( 5 \right) \cr
& = \left( {{}^5{C_3}} \right){\left( {\frac{2}{3}} \right)^3}{\left( {\frac{1}{3}} \right)^2} + \left( {{}^5{C_4}} \right){\left( {\frac{2}{3}} \right)^4}{\left( {\frac{1}{3}} \right)^1} + \left( {{}^5{C_5}} \right){\left( {\frac{2}{3}} \right)^5}{\left( {\frac{1}{3}} \right)^0} \cr
& = \frac{{10'\,\,8}}{{243}} + \frac{{5'\,\,16}}{{243}} + \frac{{1'\,\,32}}{{243}} \cr
& = \frac{{192}}{{243}} \cr} $$