An insurance company estimated that 30 % of all automobile...

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. a. What is the probability that a randomly chosen accident both was partly caused by weather conditions and involved bodily injury? b. Are the events "partly caused by weather conditions" and "involved bodily injury" independent? c. If a randomly chosen accident was partly caused by weather conditions, what is the probability that it involved bodily injury? d. What is the probability that a randomly chosen accident both was not partly caused by weather conditions and did not involve bodily injury?

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.
a. What is the probability that a randomly chosen accident both was partly caused by weather conditions and involved bodily injury?
b. Are the events "partly caused by weather conditions" and "involved bodily injury" independent?
c. If a randomly chosen accident was partly caused by weather conditions, what is the probability that it involved bodily injury?
d. What is the probability that a randomly chosen accident both was not partly caused by weather conditions and did not involve bodily injury?

Key Concepts

Intersection of Events

This concept involves finding the probability that two events occur at the same time. In probability theory, the intersection of two events is denoted by the overlap where both events occur simultaneously, and it is often calculated using the multiplication rule when one event’s probability is conditioned on the occurrence of the other.

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has occurred. It is defined as the probability of the intersection of the two events divided by the probability of the conditioning event, and is fundamental in updating probabilities when additional information is provided.

Independence

Two events are independent if the occurrence of one event does not affect the probability of the other. This is mathematically verified by checking whether the probability of their intersection equals the product of their individual probabilities. Independence is a critical concept because it simplifies the computation of joint probabilities.

Complement Rule

The complement rule deals with the probability of an event not occurring, which is the total probability of 1 minus the probability of the event occurring. This concept is essential when calculating probabilities for the absence of certain events, especially in scenarios involving multiple events or their intersections.

Transcript

00:01 So we know that the probability that an accident is do at least partly by weather is 30%.

00:06 And we know the probability of an accident being caused by bodily injury is 20%.

00:14 And then we're told that given that the accident was caused by bodily injury, that the likelihood that weather was a factor is 40%.

00:26 And so we want to know in part a, what is the probability of having weather and bodily injury, both being the intersection of those two? and so to find this, we have to, because we don't know of independence, we have to find the probability of weather times the probability of bodily injury given weather took place, or we have to find the probability of bodily injury times the probability of bodily injury, given weather took place, or we have to find the probability of bodily injury, times the probability of whether, given that bodily injury already took place.

01:01 So one of those two conditionals.

01:03 And we can do this one, because we know this probability is 0 .2, and we know that this probability conditional on the first one taking place is 0 .4.

01:15 So our answer is 0 .08.

01:19 Now, if we want, we could actually draw a picture of that circumstance, and we could say that the venn diagram, if we have weather here and bodily injury right here, we could put that 0 .08 right in here.

01:36 And then we know that these two combined together to give us the probability of bodily injury.

01:41 This would be 0 .12.

01:43 And these two together would add up to the point three.

01:46 And so that would be 0 .22.

01:48 And that means we can find what this probability of being outside here is.

01:54 As well.

01:56 But let's see what our next question is.

01:58 Our next question for part b is that we want to find our w and this independent...