Question

# The odds in favour of India winning any cricket match is $2:3$ . What is the probability that if India plays $5$ matches, it wins exactly $3$ of them?A. ${}^5{C_3}{\left( {\frac{2}{5}} \right)^2}{\left( {\frac{3}{5}} \right)^3}$B. ${}^5{C_3}{\left( {\frac{2}{5}} \right)^2}{\left( {\frac{1}{3}} \right)^3}$C. ${}^5{C_3}{\left( {\frac{2}{5}} \right)^3}{\left( {\frac{3}{5}} \right)^2}$D. ${}^5{C_3}{\left( {\frac{2}{5}} \right)^2}{\left( {\frac{1}{3}} \right)^2}$

If we look at the question, it is given that India wins out of the matches it plays, the matches it wins can be chosen ${}^5{C_3}$ ways.
Now, it is also given that the probability of India winning is $2:3$ .
India winning the match becomes $\left( {\frac{2}{5}} \right)$ and losing the match becomes$\left( {\frac{3}{5}} \right)$.
The final probability will be ${}^5{C_3}{\left( {\frac{2}{5}} \right)^3}{\left( {\frac{3}{5}} \right)^2}$