Probability of and
The probability of the and of two events, say
And with Independent Events
If events are independent then calculating the probability of and becomes simple multiplication:
If two events:
This property applies regardless of how the probabilities of
The independence principle extends to more than two
events. For
We can prove this equation by combining the definition of conditional probability and the definition of independence.
Proof: If
See the chapter on independence to learn about when you can assume that two events are independent
And with Dependent Events
Events which are not independent are called dependent events. How can you calculate the probability of the and of dependent events? If your events are mutually exclusive you might be able to use a technique called DeMorgan's law, which we cover in a later chapter. For the probability of and in dependent events there is a direct formula called the chain rule which can be directly derived from the definition of conditional probability:
The formula in the definition of conditional probability can be re-arranged to derive a general way of calculating the probability of the and of any two events:
Of course there is nothing special about
We call this formula the "chain rule." Intuitively it states that the probability of observing events