Mutually Exclusive Events
Two events, $A$ and $B$ are mutually exclusive if $P(A\cap B) = 0$
On a Venn diagram, this would be represented by two non-intersecting circles:
On a standard Venn diagram showing non-mutually events $A$ and $B$ as overlapping circles, we can represent each region via mutually exclusive events
From this diagram, the following identities can be established. $$P(A) = P(A\cap B^c) + P(A\cap B) $$ $$P(B) = (A^c\cap B) + P(A\cap B) $$ $$P(A\cup B) = P(A\cap B^c) + P(A\cap B) + (A^c\cap B)$$ $$ 1 = P(A\cup B) + P(A^c \cap B^c)$$
Independent Events
Two events, $A$ and $B$ are independent if $$P(A\cap B) = P(A)\times P(B)$$
This can also be expressed in terms of conditional probability as $$ P(A|B) = P(A) $$ this follows since $$ P(A|B) = \frac{P\cap B}{P(B)} $$ hence if A and B are independent, then $P(A\cap B) = P(A) \times P(B)$, so $$ P(A|B) = \frac{P\cap B}{P(B)} = \frac{P(A)\times P(B)}{P(B)}= P(A) $$Probability 1
A fair dice is rolled; Determine if the following events are mutually exclusive, independent or neither.
- A = "Roll a prime number"
- B = "Roll an even number"
- C = "Roll an odd number"
- D = "Roll a 1 or a 2"
solution - press button to display
The following table represents shows whether pairs of distinct events are mutually exclusive (E), independent (I) or neither (N)
A | B | C | D | |
A | --- | N | N | I |
B | N | --- | E | I |
C | N | E | --- | I |
D | I | I | I | --- |