Conditional Probability and Independence: Basics 2

We can use conditional probability to assess risks. How are the severity of an auto accident and the number of vehicles involved in the accident related?

Rule
To calculate conditional probabilities for two events A and B we use:

P(A given B) denoted P(A|B) = P(A and B)/ P(B)

P(B given A) denoted P(B|A) = P(A and B)/P(A)

These two equations can be derived from the general Both/And Rules of Unit B-2; see: Derivation Details

To investigate whether the number of vehicles involved in a crash and the type of accident are dependent events, we examine these 1995 US Motor Vehicle Crash data:

Example - Calculating Probabilities for a Single Vehicle Crash

The overall probability of a single(S) vehicle crash:
P(S) = 1891246/6613222 = 0.286

To calculate the conditional probability of a single vehicle crash given a fatal crash, we consider only fatal crashes(F):
P(S given F) = P(S and F)/ P(F)
P(S given F) = 21246/37222 = 0.571

Because the conditional probability of a single vehicle crash given a fatality is doublethe overall probability, we conclude the number of crash vehicles and the type of crash are dependent events

The overall probability of a fatal crash: P(F) = 37222/6613222 = 0.006

The same process as above can be used to calculate P(Fatal given Single). We now consider only Single Vehicle crashes:
P(F given S) = P(S and F)/ P(S)
P(F given S) == 21245/1891246 = 0.011

If we know the crash was a single vehicle, the probability of a fatality doubles.

Note: To determine whether two events are dependent, it is sufficient to consider only one of the conditional probabilities. We have done both P(Single given Fatal) and P(Fatal given Single) in these examples.

Page 2 - Determining the Conditional Probability for Two Events

Example - Calculating Probabilities for a Single Vehicle Crash