I hopefully someone can help me with this problem of conditional probability:
"A disk server receives requests from many client machines and requires 10 milliseconds to respond to each request. The probability of k additional requests in the 10-millisecond service interval is $e^{-0.9}(0.9)^{k}/k!$ for K= 0,1,2,.... If two new calls arrive while the service interval is only partially complete, what is the probability that a third new call will arrive before the server is ready to respond?"
One more thing: If it's possible, the problem should be solved using simulate with the "alias method". But I don't know the steps to make this.
thanks!
What you want the probability of three or more poisson events when given that you have at least 2 poisson events. To find this, apply the definition of conditional probability, and use the law of complements.
$$\begin{align} \mathsf P(N\ge 3\mid N\ge 2) & = \frac{1-\mathsf P(N<3)}{1-\mathsf P(N<2)} \\ & = \frac{1-(\mathsf P(N=0)+\mathsf P(N=1)+\mathsf P(N=2))}{1-(\mathsf P(N=0)+\mathsf P(N=1))} \\ & = \frac{1-e^{-0.9}(1+0.9+0.9^2/2)}{1-e^{-0.9}(1+0.9)} \\[1ex] & \approx 0.2{\small 76273\ldots} \end{align}$$
The alias method is computing algorithm used to generate a random result. I'm not sure how this would be applicable to this problem, unless you were writing code for a simulation for ... reasons beyond me.