Probability is one of the most interesting topics in mathematics and is frequently tested in many aptitude tests.
Here is a challenging probability puzzle :)
In order to win a computer tournament, a player must win at least 20 games
before he loses 5.If at any stage he loses 5 games before winning 20, then he
has lost the tournament. Given that the probability of winning a game is 1/3 for
a player , what is the probability that he will win the tournament
1/2
ReplyDeletewould it be something like:
ReplyDelete1-[(5C5*(1/3)^0*(2/3)^5)+ (6C5*(1/3)^1*(2/3)^5)+ (7C5*(1/3)^2*(2/3)^5)+………+ (24C5*(1/3)^19*(2/3)^5)] ?
so u are subtracting the probability of loss.
ReplyDeleteBut in ur case too the last match must result only in a loss. so if there are 6 matches , the last one has to be a loss , the remaining ones can be arranged
yeah i get your point....the last one has to be a loss with this logic. so i think we can put it like:
ReplyDelete1-(2/3)*[(4C4*(1/3)^0*(2/3)^4)+ (5C4*(1/3)^1*(2/3)^4)+ (6C4*(1/3)^2*(2/3)^4)+………+ (23C5*(1/3)^19*(2/3)^4)]
the tournament should not go for more than 24 games.
Well Thanks Guys for your response.
ReplyDeleteGuess Shashi Got it right
So here it goes
1)Case 1 : 0 losses , 20 wins probability = (1/3)^20
2)Case 2 : 1 losses , 20 wins (the last one is always a win) . Probability = 20C1*(2/3)*(1/3)^20
3)Case 2 : 2 losses , 20 wins P = 21C2 * (2/3)^2 *(1/3)^20
4)Case 3 : 3 losses , 20 wins P = 22C3 * (2/3)^3 * (1/3)^20
5) Case 4 : 4 losses , 20 wins , p = 23C4 * (2/3)^4 *(1/3)^20
Shashi instead subtracted the probability of getting 5 losses before 20 wins, which is also correct
However there's another interesting way to solve this problem , using the concept of a reduced sample space. Now as we know 24 matches are the maximum no of matches a player needs to play in order to win. So let us assume all the 24 were played. Now the probability is same as the probability of getting atleast 20 wins in the 24 matches. Here we don't take any restriction that the last one has to be a win .
ReplyDeleteA concept difficult to explain and understand but has been proven true! I would encourage people to try and justify this. Thanks