Question

Suppose that on a certain stretch of highway, cars pass at an average rate of five cars per minute. Assume that the duration of time between successive cars follows the exponential distribution.

a. On average, how many seconds elapse between two successive cars?

b. After a car passes by, how long on average will it take for another seven cars to pass by?

c. Find the probability that after a car passes by, the next car will pass within the next 20 seconds.

d. Find the probability that after a car passes by, the next car will not pass for at least another 15 seconds.

Short answer

1. The seconds will elapse to pass a car will be $12$
   
2. The seconds will elapse to pass $7$car will be $84$.
3. The probability that after a car passes by, the next car will pass within the next $20$seconds will be $0.8111$.
4. The probability that after a car passes by, the next car will not pass for at least another $15$seconds will be$0.2865$.

Step-by-step solution

Part (a) Step 1: Given information 

On a certain stretch of highway, cars pass at an average rate of five cars per minute.

The duration of time between successive cars follows the exponential distribution.

Part (a) Step 2: Solution 

It is shown that on average $5$cars pass in $1$minute or in $60$seconds. So, number of seconds elapsed to pass $1$car will be:

$=\frac{60}{5}$

$=12$

Part (b) Step 1: Given information 

On a certain stretch of highway, cars pass at an average rate of five cars per minute.

The duration of time between successive cars follows the exponential distribution.

Part (b) Step 2: Solution 

It has been calculated in part $\left(a\right)$that it takes $12$seconds to pass a car, so it will take the following seconds to pass another $7$cars:

$=12\times 7$

$=84$

Part (c) Step 1: Given information

On a certain stretch of highway, cars pass at an average rate of five cars per minute.

The duration of time between successive cars follows the exponential distribution.

Part (c) Step 2: Solution

From the part (a), it is clear that the random variable X will follow exponential distribution as below:

$X~\mathrm{Exp}\left(12\right)$

Here,
$\mu =12$

$m=\frac{1}{12}$

As a result, the exponential distribution's generic probability distribution function is:

$P(X\le x)=1-e^{-mx}$

As a result, the necessary probability can be computed as follows:

$P(x\le 20)=1-e^{-\frac{1}{12}\times 20}$

$=1-0.1889$

$=0.8111$

Part (d) Step 1: Given information 

On a certain stretch of highway, cars pass at an average rate of five cars per minute.

The duration of time between successive cars follows the exponential distribution.

Part (d) Step 2: Solution (Part d)

Here, the required probability can be calculated as below:

$P(x>15)=1-\left(1-e^{-\frac{1}{12}\times 15}\right)$

$=e^{-\frac{1}{12}\times 15}$

$=0.2865$