Question

c) An insurance company estimated that 30% of all automobile accidents were partly caused by
weather conditions and that 20% of all automobile accidents involved bodily injury. Further, of
those accidents that involved bodily injury, 40% were partly caused by weather conditions.
I. What is the probability that a randomly chosen accident both was partly caused by
weather conditions and involved bodily injury?
II. Are the events “partly caused by weather conditions” and “involved bodily injury”
independent?
III. If a randomly chosen accident was partly caused by weather conditions, what is the
probability that it involved bodily injury?
IV. What is the probability that a randomly chosen accident both was not partly caused by
weather conditions and did not involve bodily injury?

Answer 1

i. Suppose that $x$ is a total number of accidents. $0.2x$ is a number of accidents involving bodily injury. $0.2x\cdot0.4=0.08x$ is a number of accidents involving bodily injury that were caused by weather conditions. Thus, the probability is 0.08.

ii. The following condition: $P(A\cap B)=P(A)P(B)$ is equivalent to independency of events $A$ and $B$ (see [1, page 125]). Event $A$ : The accident was partly caused by weather conditions; Event $B$: The accident involved bodily injury. $P(A\cap B)=0.2\cdot0.4=0.08,$ whereas, $P(A)P(B)=0.2\cdot0.3=0.06$. Thus, the events are not independent.

iii. Our aim is to compute $P(B|A)$ . We remind that (see [1, page 115])

$P(B|A)=\frac{P(A\cap B)}{P(A)}=\frac{0.2\cdot0.4}{0.3}=0.2667$.

iv. We know that 30% of accidents were partly caused by weather conditions and 20% of accidents involved bodily injuries. 8% $(0.2\cdot0.8\cdot100\%)$ of accidents were partly caused by weather conditions and involved bodily injuries. Thus, $100\%-30\%-20\%+8\%=58\%$ of accidents were not partly caused by weather conditions and did not involve bodily injuries. Thus, the respective probability is 0.58

References:

[1] Feller, W. (1967). An Introduction to Probability Theory and Its Applications (Third ed.). New York: Wiley.

Answers: i. 0.08; ii. The events are not independent; iii. 0.2667; iv. 0.58.