An old house in Pomona, CA is inhabited by a variety of ghosts. Ghost appearances occur in the house according to a Poisson process having a rate of 1.4 ghosts per hour. A professor from Cal Poly Pomona has developed a device that can be used to detect ghost appearances. Suppose it is now 1:00 p.m. and the last ghost appearance (the 6th overall) was at 12:35 p.m.
What is the probability that the 7th ghost will appear before 1:30 p.m., to the nearest three decimal places?

If the ghost appearances occur in the house according to a Poisson process with mean m, the time between appearances follows a exponential distribution with mean 1/m. so, the probability that the next ghost appearance happens before x hours is equal to:

[tex]P(X\leq x)=1-e^{-xm}[/tex]

Then, replacing m by 1.4 ghosts per hour we get:

[tex]P(X\leq x)=1-e^{-1.4x}[/tex]

Additionally, The exponential distribution have a memoryless property, so if it is now 1:00 p.m. and we want the probability that ghost appear before 1:30 p.m., we need to find the difference in hours from 1:00 p.m and 1:30 p.m. no matter that the last ghost appearance was at 12:35 p.m.

Therefore, there are 0.5 hours between 1:00 p.m. and 1:30 p.m, so the probability that the 7th ghost will appear before 1:30 p.m is calculated as:

[tex]P(x\leq 0.5)=1-e^{-1.4*0.5} =0.503[/tex]