Answer:
35.6 seconds
Step-by-step explanation:
The Poisson distribution has a probability model that follows the given relationship:
[tex]P(X=x) = \frac{\lambda^xe^{-\lambda}}{x!}[/tex]
For x = zero calls at a probability of 50%, the arrival rate during the call is:
[tex]P(X=0)=0.50 = \frac{\lambda^0e^{-\lambda}}{0!}\\e^{-\lambda}=0.50\\\lambda=0.6931\ cars/ n\ seconds[/tex]
Since cars arrive at a rate of 70 per hour, the length of call 'n' in seconds is:
[tex]r = \frac{70}{h}*\frac{1h}{3,600s} = 0.019444\ cars/s\\n= \frac{\lambda}{r}=\frac{0.6931}{0.019444} =35.6\ seconds[/tex]
The call should be at most 35.6 seconds long to ensure a 50% probability that no cars arrive during the call.
