Condition 2 (exhaustive): , so
Total Law of Property expressed in conditional property:
Example: Binary Communication System
There is a probability of sending a 0, and probability of sending a 1. If the bit sent is a 0, there is 0.9 that the output is correct, 0.1 probability of flipped bit. Else if the bit sent is a 1, there is 0.8 chance that that output is correct, but 0.2 chance that the bit is flipped.
To find the probability of error, we add the total probability of each error up (total law property)
Given , find:
Family has two children, and
We want to find
Since they are all equally likely, the probability of each one happening is
And we can know and find the conditional probability.
There are people in the room, we want the probability of at least 2 people having same birthday.
The total possibilities of date of birth is days. and is number of possibilities with constrain that no people share birthdays.