Question

The probability that Sania wims Wimbledon tournaments final is $\frac{1}{3}$. If Sania Mirza plays $3$ round of Wimbledon final, the probability that she wins at least one round of match is:

• A. $\frac{8}{27}$
• B. $\frac{19}{27}$
• C. $\frac{10}{27}$
• D. $\frac{17}{27}$

Answer 1

The correct option is B $\frac{19}{27}$

Probability that Sania wins Wins Wimbledon tournaments final $=\frac{1}{2}$

$\Rightarrow$ Probability that Sania losses $=\frac{2}{3}$

$\Rightarrow$ i.e, $p=\frac{1}{3}$ $q=\frac{2}{3}$

$\Rightarrow$ Total number of rounds, $n=3$

$\Rightarrow$ By Bernoulli's theorm, $P={}^n{C}_r{p}^r{q}^{n-r}$

P (She wins atleast $1$ round )

$={}^3{C}_1{\left(\frac{1}{3}\right)}^1{\left(\frac{2}{3}\right)}^2+{}^3{C}_2{\left(\frac{1}{3}\right)}^2\left(\frac{2}{3}\right)+{}^3{C}_3{\left(\frac{1}{3}\right)}^3{\left(\frac{2}{3}\right)}^0$

$=\frac{3\times 4}{27}+\frac{3\times 2}{27}+\frac{1}{27}$

$=\frac{12+6+1}{27}$

$=\frac{19}{27}$

Hence, the answer is $\frac{19}{27}.$