Question

The probability that Dhoni will hit a century in every ODI match he plays is $\dfrac{1}{5}$. If he plays 6 matches in World cup 2011, the probability that he will score century in all the 6 matches is:
A. $\dfrac{1}{{3125}}$
B. $\dfrac{5}{{3125}}$
C. $\dfrac{{4096}}{{15625}}$
D. $\dfrac{{1555}}{{3125}}$

Answer 1

Hint: We will use Bernoulli's theorem as there are only possible outcomes, either Dhoni will hit century or he will not score a century in an ODI match. From the given probability of scoring a century, calculate the probability of not scoring a century. Then substitute the values in Bernoulli's theorem is $^n{C_r}{p^r}{q^{n - r}}$ , to get the required probability.

Complete step-by-step answer:
Let the probability of getting a century in an ODI match be$p$ which is given as $\dfrac{1}{5}$.
Now we will calculate the probability of not hitting a century in an ODI match by subtracting the probability of hitting a century from 1.
Let us denote the above probability by $q$.
$
   \Rightarrow q = 1 - p \\
   \Rightarrow q = 1 - \dfrac{1}{5} \\
   \Rightarrow q = \dfrac{4}{5} \\
$
We have to find the probability of scoring a century in all 6 matches.
We will Bernoulli's theorem to find the required probability.
Bernoulli's theorem is $^n{C_r}{p^r}{q^{n - r}}$ , where $n$ is the total number of matches, $r$ is the required number of matches, $p$ is the probability of success and $q$ is the probability of failure.
On substituting the values for total and required matches as 6, we will get,
$
  { \Rightarrow ^6}{C_6}{\left( {\dfrac{1}{5}} \right)^6}{\left( {\dfrac{4}{5}} \right)^{6 - 6}} \\
  { \Rightarrow ^6}{C_6}{\left( {\dfrac{1}{5}} \right)^6} \\
$
Now, we know that $^n{C_n} = 1$
Therefore, the required probability is ${\left( {\dfrac{1}{5}} \right)^6} = \dfrac{1}{{15625}}$
Hence the required probability is $\dfrac{1}{{15625}}.$


Note: The sum of the probability of success and probability of failure is equal to 1. Also, the probability of any event cannot be less than 0 and cannot be greater than 1.