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Question: The probability that Dhoni will hit a century in every ODI match he lays is . If he plays matches in...

The probability that Dhoni will hit a century in every ODI match he lays is . If he plays matches in World Cup 2011 , the probability that he will score century in all the test matches is :15\dfrac{1}{5}
A 13125\dfrac{1}{{3125}}
B 53125\dfrac{5}{{3125}}
C 409615625\dfrac{{4096}}{{15625}}
D 15553125\dfrac{{1555}}{{3125}}

Explanation

Solution

In this question use the Bernoulli's trials , So from Bernoulli trial Probability = nCrprqnr{}^n{C_r}{{\text{p}}^r}{{\text{q}}^{n - r}} where n is number of trial and r will be the success getting out of n hence for this nn = 66 r=6r = 6 p=15{\text{p}} = \dfrac{1}{5} q=45{\text{q}} = \dfrac{4}{5} from this we will proceed towards answer .

Complete step-by-step answer:
As in the given question the probability that Dhoni will hit century in every ODI matches , is 15\dfrac{1}{5} we have to find that the probability that he will score century in all the 66 test matches ,
So the probability of getting a century , p=15{\text{p}} = \dfrac{1}{5}
Probability of not getting a century , q=45{\text{q}} = \dfrac{4}{5}
And it is given that the total number of matches is 66
So in this we use Bernoulli's trials as
Bernoulli trials approaches the probability of success as the number of trials increases towards infinity ,
So from Bernoulli trial
Probability = nCrprqnr{}^n{C_r}{{\text{p}}^r}{{\text{q}}^{n - r}} where n is number of trial and r will be the success getting out off n ,
In this case nn = 66 because 66 test matches is played and out which we want success in all so r=6r = 6
And from above we know that the p=15{\text{p}} = \dfrac{1}{5} and q=45{\text{q}} = \dfrac{4}{5} so on putting these values in Bernoulli trial
Probability that he will hit century in all matches is = 6C6(15)6(45)66{}^6{C_6}{\left( {\dfrac{1}{5}} \right)^6}{\left( {\dfrac{4}{5}} \right)^{6 - 6}}
we know that the value of 6C6=1{}^6{C_6} = 1 and 56=3125{5^6} = 3125 on solving ,
Probability = 13125\dfrac{1}{{3125}}
Probability that he will score century in all the 66 test matches is 13125\dfrac{1}{{3125}} hence Option A is correct .

Note: If in this question it will asked that the probability he will not score century in all the 66 test matches then we will use Probability of unsuccessful event = 11 - Probability of successful event , and Probability of successful event = 13125\dfrac{1}{{3125}} or we will also put in nCrprqnr{}^n{C_r}{{\text{p}}^r}{{\text{q}}^{n - r}} r=0r = 0 in this equation .