A study of drinking and driving has found that 40% of all fatal auto accidents are attributed to drunk drivers, 2% of all auto accidents are fatal, and drunk drivers are responsible for 20% of all accidents. Find the percentage of non-fatal accidents caused by drivers who are not drunk.

#### Solution

Let $D$ is the subset attributable to drunk drivers, and $F$ is the subset that are fatal.

Given that $P(D|F) = 0.40$, $P(F) = 0.02$, $P(D) = 0.20$.

So,

` $$ \begin{aligned} P(D|F) &= \frac{P(D\cap F)}{P(F)}\\ \Rightarrow P(D \cap F) &= P(D|F) P(F)\\ &= 0.40*0.02\\ &= 0.008 \end{aligned} $$ `

We wish to find the probability of non-fatal $F^\prime$ accidents caused by drivers who are not drunk $D^\prime$, that is

` $$ \begin{aligned} P(D^\prime|F^\prime) &= \frac{P(D^\prime \cap F^\prime)}{P(F^\prime)} \\ &= \frac{1-P(D\cup F)}{1-P(F)} \\ &= \frac{1-(P(D) +P(F) -P(D\cap F))}{1-P(F)}\\ &= \frac{1-(0.20+0.02 - 0.008)}{1-0.02}\\ &=\frac{1-0.212}{0.98}= 0.804. \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators