Answer:
The proportion of cars can get through the toll booth in less than 3 minutes is 67%.
Step-by-step explanation:
Let the random variable <em>X</em> be defined as the service times at a tool booth.
The random variable <em>X</em> follows an Exponential distribution with parameter <em>μ</em> = 2.7 minutes.
The probability density function of <em>X</em> is:

Compute the probability that a car can get through the toll booth in less than 3 minutes as follows:

![=\frac{1}{2.7}\cdot \int\limits^{3}_{0} {e^{-x/2.7}} \, dx \\\\=\frac{1}{2.7}\cdot [-\frac{e^{-x/2.7}}{1/2.7}]^{3}_{0}\\\\=1-e^{-3/2.7}\\\\=0.6708](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B2.7%7D%5Ccdot%20%5Cint%5Climits%5E%7B3%7D_%7B0%7D%20%7Be%5E%7B-x%2F2.7%7D%7D%20%5C%2C%20dx%20%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B2.7%7D%5Ccdot%20%5B-%5Cfrac%7Be%5E%7B-x%2F2.7%7D%7D%7B1%2F2.7%7D%5D%5E%7B3%7D_%7B0%7D%5C%5C%5C%5C%3D1-e%5E%7B-3%2F2.7%7D%5C%5C%5C%5C%3D0.6708)
Thus, the proportion of cars can get through the toll booth in less than 3 minutes is 67%.