Summary
Highlights
Mutually exclusive events are defined as events that cannot occur together or happen at the same time. Visually, a Venn diagram shows a clear separation between these events with no intersection. Examples include male/female, pass/fail, and alive/dead. If event A is getting an even number on a die roll (2, 4, 6) and event B is getting an odd number (1, 3, 5), A and B are mutually exclusive because they have no similar elements.
For events A (even numbers: 2, 4, 6) and C (numbers less than 5: 1, 2, 3, 4) from a die roll, they are not mutually exclusive. This is because they share common elements like 2 and 4, meaning there is an intersection in their Venn diagram representation.
Independent events are events where the occurrence of one does not affect the occurrence of another. Mathematically, the conditional probability of B given A (P(B|A)) is equal to the probability of B (P(B)). Similarly, P(A|B) = P(A). This means knowing that one event has happened does not change the probability of the other event.
Dependent events are events where the occurrence of one significantly alters the probability of the other. In this case, P(B|A) is not equal to P(B). An example involving survey data comparing 'female' and 'in favor' shows that these events are dependent. The probability of being female given 'in favor' (0.2105) is not equal to the overall probability of being female (0.4), indicating dependence.
To further confirm dependence, one can also check if P(in favor|female) equals P(in favor). In the example, P(in favor|female) is calculated as 0.1, while P(in favor) is 0.19. Since these values are not equal, it reconfirms that the events 'female' and 'in favor' are dependent.
The complement of an event A (denoted as A' or A bar) represents all outcomes of an experiment that are not in A. It is the 'totally opposite' of A. The sum of the probability of an event and its complement always equals 1 (P(A) + P(A') = 1). An example discusses taxpayers: if 4,000 out of 20,000 have been audited (event A), the probability of A is 0.2. Its complementary event (A') is taxpayers who have never been audited, and its probability is 1 - 0.2 = 0.8.