Conditional Probability: Independent Events
If the probability of an event is subject to a restriction on the sample space, the probability is said to be conditional. Conditional probability is the probability of the occurrence of an event, say A, subject to the occurrence of a previous event, say B. We define the conditional probability of event A, given that B has occurred, in case of A and B being independent events, as the probability of event A.
P(A|B) = P(A)
It is so because independent events are those whose probabilities are in no way affected by the occurrence of each other.
Let us take the same true-false test. As the success answers are independent of each other we can say that the probability of success of the second answer given that the first answer is a success is simply the probability of the success of the second answer, i.e.
P(S2|S1) = P(S2) = 0.5