Probability of at least one event There are 2 independent events, probability that Exam A is a success is 0.4. Probability that Exam B is a success is 0.7. What is the probability that at least one of these is a success.
So I thought the way 'at-least- one of these is a success is 'a is a success' or 'b is a success' or 'a and b is a success'. But the answer is  'a is a success' or 'b is a success' - 'a and b is a success'. I am confused as to why this is.
 A: The probability of at least one of the events occurring is the total probability minus the probability that none occur.
In this case, use $1-$(Prob. of failed A)$\times$(Prob. of failed B)
This is equal to $1-(0.6)\times(0.3)=1-0.18=0.82$ or $82$%
A: You could think of at least one of these is a success is any of 


*

*a is a success and b is a failure

*b is a success and a is a failure

*a and b are each a success
and add up the probabilities.  That will give you the correct answer.
You could note that the probability that a is a success and b is a failure or a and b are each a success is equal to the probability that a is a success, and similarly  the probability that b is a success and a is a failure or a and b are each a success is equal to the probability that b is a success.
Then adding together the probability that a is a success and the probability that b is a success will count the probability that a and b are each a success  twice.  So to get the correct answer, you will need to subtract the probability that a and b are each a success.
This is called the principle of inclusion-exclusion.
