Putting It Together: Techniques of Integration

Traffic Accidents in a City

This is a picture of a city street with a traffic signal. The picture has very busy lanes of traffic in both directions.

In the chapter opener, we stated the following problem: Suppose that at a busy intersection, traffic accidents occur at an average rate of one every three months. After residents complained, changes were made to the traffic lights at the intersection. It has now been eight months since the changes were made and there have been no accidents. Were the changes effective or is the 8-month interval without an accident a result of chance?

Probability theory tells us that if the average time between events is $k$ , the that $X$ , the time between events, is between $a$ and $b$ is given by

P (a \leq x \leq b) = \int_{a}^{b} f (x) d x where f (x) = {\begin{matrix} 0 if x < 0 k e^{- k x} if x \geq 0 \end{matrix}

$P\left(a\le x\le b\right)={\displaystyle\int }_{a}^{b}f\left(x\right)dx\text{ where }f\left(x\right)= \begin{cases} 0\text{ if }x<0 \\ k{e}^{-kx}\text{ if }x\ge 0\end{cases}$

Thus, if accidents are occurring at a rate of one every 3 months, then the probability that $X$ , the time between accidents, is between $a$ and $b$ is given by

P (a \leq x \leq b) = \int_{a}^{b} f (x) d x where f (x) = {\begin{matrix} 0 if x < 0 3 e^{- 3 x} if x \geq 0 \end{matrix}

$P\left(a\le x\le b\right)={\displaystyle\int }_{a}^{b}f\left(x\right)dx\text{ where }f\left(x\right)= \begin{cases} 0\text{ if }x<0 \\ 3{e}^{-3x}\text{ if }x\ge 0\end{cases}$

To answer the question, we must compute $P\left(X\ge 8\right)={\displaystyle\int }_{8}^{+\infty }3{e}^{-3x}dx$ and decide whether it is likely that 8 months could have passed without an accident if there had been no improvement in the traffic situation.

Show Solution

We need to calculate the probability as an improper integral:

\begin{matrix} P (X \geq 8) & = \int_{8}^{+ \infty} 3 e^{- 3 x} d x = lim t \to + \infty \int_{8}^{t} 3 e^{- 3 x} d x = lim t \to + \infty {- e^{- 3 x} |}_{8}^{t} = lim t \to + \infty (- e^{- 3 t} + e^{- 24}) \approx 3.8 \times 10^{- 11} . \end{matrix}

$\begin{array}{cc}\hfill P\left(X\ge 8\right)& ={\displaystyle\int }_{8}^{+\infty }3{e}^{-3x}dx\hfill \\ & =\underset{t\to \text{+}\infty }{\text{lim}}{\displaystyle\int }_{8}^{t}3{e}^{-3x}dx\hfill \\ & =\underset{t\to \text{+}\infty }{\text{lim}}{\text{-}{e}^{-3x}|}_{8}^{t}\hfill \\ & =\underset{t\to \text{+}\infty }{\text{lim}}\left(\text{-}{e}^{-3t}+{e}^{-24}\right)\hfill \\ & \approx 3.8\times {10}^{-11}.\hfill \end{array}$

The value $3.8\times {10}^{-11}$ represents the probability of no accidents in 8 months under the initial conditions. Since this value is very, very small, it is reasonable to conclude the changes were effective.

Module 3: Techniques of Integration

Putting It Together: Techniques of Integration

Traffic Accidents in a City

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