The probability that Sania wins Wimbledon tournaments final is \[\dfrac{1}{3}\]. If Sania Mirza plays 3 round of Wimbledon final, the probability that she losses all the rounds are:
A. \[\dfrac{8}{{27}}\]
B. \[\dfrac{{19}}{{27}}\]
C. \[\dfrac{{10}}{{27}}\]
D. \[\dfrac{{17}}{{27}}\]

Hint: We will first consider the given data that the probability of winning is \[\dfrac{1}{3}\] and we need to find the probability if she loses all the three 3 rounds. So, we will first find the probability that Sania loses one round. Now, we will use the concept of Bernoulli’s theorem, to determine the probability when she loses all the 3 rounds. Hence, we will get the desired result.

Complete step by step answer:

We will first consider that the probability of winning is \[p = \dfrac{1}{3}\].
From this, we can find the probability of losing by subtracting the winning probability from 1.
Thus, we get,
\[
   \Rightarrow q = 1 - p \\
   \Rightarrow q = 1 - \dfrac{1}{3} \\
   \Rightarrow q = \dfrac{2}{3} \\
 \]
Hence, the probability of losing is \[\dfrac{2}{3}\].
There is a total number of rounds given by \[n = 3\].
We have to find the probability that she loses all the rounds so, we will use the Bernoulli’s theorem which states that \[P = {}^n{C_r}{p^r}{q^{n - r}}\] in finding this, where \[\dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
Also, we will take \[r = 0\] as we have to find the probability of losing all the rounds.
Thus, we have,
\[
   \Rightarrow P = {}^3{C_0}{\left( {\dfrac{1}{3}} \right)^0}{\left( {\dfrac{2}{3}} \right)^{3 - 0}} \\
   \Rightarrow P = \dfrac{{3!}}{{0!3!}}\left( 1 \right){\left( {\dfrac{2}{3}} \right)^3} \\
   \Rightarrow P = \dfrac{8}{{27}} \\
 \]
Hence, we can conclude that the probability of losing all the rounds is \[\dfrac{8}{{27}}\].
Thus, option A is correct.

Note: We have used the Bernoulli’s concept as we are given the probability of winning and we have found for the probability of losing and the number of rounds are given so, this theorem fits up here perfectly. The expression \[{}^n{C_r}\] can be found by solving it \[\dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]. The probability of losing can easily be determined by subtracting the probability of winning from 1. Substitution should be done properly in the formula. Remember to take \[r = 0\] as we have found the probability of losing all the rounds.