Question

The probability that India wins a cricket test match against England is $\dfrac{1}{3}$ . If India and england plays 3 matches, the probability that India will win at least one match is
A) $\dfrac{8}{{27}}$
B) $\dfrac{{19}}{{27}}$
C) $\dfrac{1}{{27}}$
D) $\dfrac{9}{{27}}$

Answer 1

Hint: Try to find the total probability that india will not win any test match then use that probability to find that india will win at least 1 match by subtracting it from 1.
Complete step by step answer:
 It is given that the probability that india will win no game at all will be given by
$1 - \dfrac{1}{3} = \dfrac{{3 - 1}}{3} = \dfrac{2}{3}$
\[\therefore P(Los{s_{India}}) = \dfrac{2}{3}\]
It is given that 3 consecutive games are played therefore the probability that india will lose all of them will be \[\therefore P(Los{s_{India}}) = \dfrac{2}{3} \times \dfrac{2}{3} \times \dfrac{2}{3} = \dfrac{8}{{27}}\]
Now for getting the required result that India won at least 1 match will be given by
\[P(Wi{n_{India}}) = 1 - \dfrac{8}{{27}} = \dfrac{{27 - 8}}{{27}} = \dfrac{{19}}{{27}}\]
So our required answer is \[\dfrac{{19}}{{27}}\]
Therefore, option B is the correct answer.

Note: The highest probability can only be 1 if in any case you get a probability that is greater than 1 that means you made a mistake somewhere and also we can use this 1 to get a complement of probability we need.