- #1
Hiero
- 322
- 68
Homework Statement
In a short time dt, the probability that a car passes an observer is dt/τ. In a random time interval of length T, what is the probability P(n) that exactly n cars pass the observer?
The Attempt at a Solution
I can only see how to find P(0) the probability that no cars pass the observer.
To do this I treat successive time intervals of length dt as discrete events then take the limit as dt goes to zero:
The probability that no cars pass in dt is (1-dt/τ) so the probability that no cars pass in a time T = n•dt should be (1-dt/τ)n = (1-dt/τ)T/dt
The limit as dt→0 involves the famous limit for Euler’s number, and so we get:
P(0) = e-T/τ
We can say that one minus this is the probability of at least one car passing, i.e.:
1 - P(0) = Σn = 1 to ∞[P(n)]
But I am at a loss as to how to find the probability for a particular n.
Thanks.