You are throwing darts at a dart board, aiming at the center. The second dart hit the board farther from the center than the first. What is the probability the third dart will also hit the board farther from the center than the first? Assume that all the throws are independent.
The trick here is to think in terms of joint distribution than in terms of the cdf P(x>r). The model for distribution P(x>r) is not known, it could be anything but for the joint we know that the throws are independent, therefore the 3! possibilities are equiprobable. Conditioning on the fact that X_2>X_1, we are left with only 3 possibilities:
ReplyDelete(X_3> X_2>X_1), (X_2>X_3>X_1) and (X_2>X_1>X_3) of which 2 are favorable. Thus ans is 2/3