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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. ${}^5{C_3}{\left( {\frac{2}{5}} \right)^2}{\left( {\frac{1}{3}} \right)^2}$

Last updated date: 19th Jul 2024
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Hint: First find out the ways in which the matches are won by India, then use the ratio given to find the probability of India winning and losing the match and then apply these values to find the final probability.

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 ${}^5{C_3}{\left( {\frac{2}{5}} \right)^3}{\left( {\frac{3}{5}} \right)^2}$
Note: In this question, we have been given that India wins the match 3 out of 5 times, therefore the total ways we can select a match is calculated first, then using the given probability we found out the final probability.