Probability can be studied in conjunction with set theory, with Venn Diagrams being particularly useful in analysis. The probability of a certain event occurring, for example, can be represented by P(A). The probability of a different event occurring can be written P(B). Clearly, therefore, for two events A and B,
P(A) + P(B) - P(AnB) = P(AuB)
Events A and B are mutually exclusive if they have no events in common. In other words, if A occurs B cannot occur and vise versa. If, for example, we are asked to pick a card from a pack of 52, the probability that the card is red is 1/2 . The probability that the card is a club is 1/4 . However, if the card is red it can't be a club. These events are therefore mutually exclusive. If two events are mutually exclusive, P(AnB) = 0, so P(A) + P(B) = P(AuB)
Two events are independent if the first one does not influence the second. For example, if a bag contains 2 blue balls and 2 red balls and two balls are selected randomly, the events are:
a) independent if the first ball is replaced after being selected
b) not independent if the first ball is removed without being replaced. In this instance, the are only three balls remaining in the bag so the probabilities of selecting the various colours have changed.
Two events are independent if (and only if): P(AnB) = P(A)P(B) .
Conditional probability is the probability of an event occurring, given that another event has occurred. For example, the probability of John doing mathematics at A-Level, given that he is doing physics may be quite high. P(A|B) means the probability of A occurring, given that B has occurred. For two events A and B,
P(AnB) = P(A|B)P(B) and similarly P(AnB) = P(B|A)P(A)