Answer
Verified
401.4k+ views
Hint: It is possible that X will win a match of first n games or X will lose the match or the match will draw but here we had to solve the probability that X will win a match so let the probability of winning the match be P. And we all know that \[{(^n}{C_r}) = \dfrac{{n!}}{{r!}}(n - r)!\].
Complete step-by-step answer:
It is clear that X will win the match after the ( n + 1 ) game.
And this is possible by two mutually exclusive ways and these 2 ways also have three cases.
( 1 ) The 1st way of winning the match is \[{P_1}\].
Case (a) X wins exactly one of the first n games
Case (b) draw ( n – 1 ) games
Case (c) wins the ( n + 1 ) the games.
So, the probability of this way is \[{(^n}{P_1}a{b^{n - 1}})a\].
( 2 ) Now let the 2nd way of winning the match is \[{P_2}\].
Case (a) X loses exactly one of the first n games, wins exactly one of the first n games
Case (b) draws ( n – 2 ) games.
Case (c) wins the ( n + 1 ) game.
So, the probability of this way is \[{(^n}{P_2})(ac){b^{n - 2}}a\].
Now the probability X wins the match after ( n + 1 ) game is
P (X) = \[{P_1} + {P_2}\]
Now, put the value of \[{P_1}\] and \[{P_2}\] in the above equation.
P ( X ) = \[{(^n}{P_1}a{b^{n - 1}})a\]+ \[{(^n}{P_2})(ac){b^{n - 2}}a\] \[\left[ {{(^n}{P_2}) = n(n - 1)} \right]\]
P ( X ) = \[n{a^2}{b^{n - 1}} + \;n(n - 1){a^2}{b^{n - 2}}c\]
P ( X ) = \[{a^2}\left[ {n{b^{n - 1}} + \;n(n - 1){b^{n - 2}}c} \right]\]
So, the probability X wins the match after ( n + 1 ) games.
Hence B is the correct option.
Note: whenever we come up with this type of problem then we must focus on all the possible ways that can be made by the given statement. And then we have to find the probability for all the possible cases and in the end we have to add up all these to give us the desired result.
Complete step-by-step answer:
It is clear that X will win the match after the ( n + 1 ) game.
And this is possible by two mutually exclusive ways and these 2 ways also have three cases.
( 1 ) The 1st way of winning the match is \[{P_1}\].
Case (a) X wins exactly one of the first n games
Case (b) draw ( n – 1 ) games
Case (c) wins the ( n + 1 ) the games.
So, the probability of this way is \[{(^n}{P_1}a{b^{n - 1}})a\].
( 2 ) Now let the 2nd way of winning the match is \[{P_2}\].
Case (a) X loses exactly one of the first n games, wins exactly one of the first n games
Case (b) draws ( n – 2 ) games.
Case (c) wins the ( n + 1 ) game.
So, the probability of this way is \[{(^n}{P_2})(ac){b^{n - 2}}a\].
Now the probability X wins the match after ( n + 1 ) game is
P (X) = \[{P_1} + {P_2}\]
Now, put the value of \[{P_1}\] and \[{P_2}\] in the above equation.
P ( X ) = \[{(^n}{P_1}a{b^{n - 1}})a\]+ \[{(^n}{P_2})(ac){b^{n - 2}}a\] \[\left[ {{(^n}{P_2}) = n(n - 1)} \right]\]
P ( X ) = \[n{a^2}{b^{n - 1}} + \;n(n - 1){a^2}{b^{n - 2}}c\]
P ( X ) = \[{a^2}\left[ {n{b^{n - 1}} + \;n(n - 1){b^{n - 2}}c} \right]\]
So, the probability X wins the match after ( n + 1 ) games.
Hence B is the correct option.
Note: whenever we come up with this type of problem then we must focus on all the possible ways that can be made by the given statement. And then we have to find the probability for all the possible cases and in the end we have to add up all these to give us the desired result.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE