Consider a Poisson arrival process with rate λ per hour. To simplify notation, we let a=P(0,1), b=P(1,1), and c=P(2,1), where P(k,1) is the probability of exactly k arrivals over an hour-long time interval. What is the probability that we will have “at most one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 and 12:00"? Your answer should be an algebraic function of a, b, and c in standard notation.
(a*c)+(b^2)
thats right
not sure of the ans. , anyone open to discussing
Well, well, well! We have a fun probability question here!
Let's break it down, shall we? We want the probability of having at most one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 and 12:00.
So, let's start with the probability of having at most one arrival between 10:00 and 11:00, which is P(0,1) + P(1,1) (since we want the probability of 0 or 1 arrival in that hour).
Now, for the probability of exactly two arrivals between 10:00 and 12:00, we need to consider what could happen between 11:00 and 12:00. It can either be 0 arrivals (P(0,1)), 1 arrival (P(1,1)), or 2 arrivals (P(2,1)).
So, the overall probability should be the product of the probabilities of these two events: (P(0,1) + P(1,1)) * (P(0,1) + P(1,1) + P(2,1)).
We can simplify this expression as (a + b) * (a + b + c).
So, there you have it! The probability is (a + b) * (a + b + c).
Hope that puts a smile on your face!
To find the probability that there will be at most one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 and 12:00, we can break it down into two cases:
Case 1: At most one arrival between 10:00 and 11:00:
This means either no arrivals or one arrival between 10:00 and 11:00. So, the probability for this case can be calculated as:
P(0,1) + P(1,1) = a + b
Case 2: Exactly two arrivals between 10:00 and 12:00:
This means two arrivals between 10:00 and 12:00, but at most one of them should have occurred between 10:00 and 11:00. We can calculate this probability as the product of two probabilities:
P(1,1) * P(1,1) = b * b
To obtain the total probability, we need to calculate the probabilities for both cases and then add them together:
P(total) = P(Case 1) + P(Case 2)
= a + b + b * b
Therefore, the probability that we will have at most one arrival between 10:00 and 11:00 and exactly two arrivals between 10:00 and 12:00 is given by the algebraic function:
P(total) = a + b + b^2