Question

Speeding linked to fatal car crashes. According to the National Highway Traffic and Safety Administration’s National Center for Statistics and Analysis (NCSA), “Speeding is one of the most prevalent factors contributing to fatal traffic crashes” (NHTSA Technical Report, August 2005). The probability that speeding is a cause of a fatal crash is .3. Furthermore, the probability that speeding and missing a curve are causes of a fatal crash is .12. Given speeding is a cause of a fatal crash, what is the probability that the crash occurred on a curve?

Short answer

The probability is 0.4.

Step-by-step solution

Important formula

The formula is $\mathbf{P}\mathbf{(}\mathbf{A}\cap \mathbf{B}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{)}\mathbf{.}\mathbf{P}\mathbf{\left(}\mathbf{B}\mathbf{\right|}\mathbf{A}\mathbf{)}$.

The probability that the crash occurred on a curve.

Here, $\mathrm{P}\left(\mathrm{G}\right)=0.3=$Speeding is a cause of a fatal crash.

$\mathrm{P}\left(\mathrm{F}|\mathrm{G}\right)=0.12=$Speeding and missing a curve are causes of a fatal crash.

$\begin{array}{c}\mathrm{P}(\mathrm{G}\cap \mathrm{F})=\mathrm{P}\left(\mathrm{G}\right)\times \mathrm{P}\left(\mathrm{F}\right|\mathrm{G})\\ 0.12=\left(0.38\right)\mathrm{P}\left(\mathrm{F}\right|\mathrm{G})\\ \mathrm{P}\left(\mathrm{F}\right|\mathrm{G})=0.4\end{array}$

Therefore the probability is 0.4.