For two events and , the conditional probability of event , is the probability of event given the occurence (or non-occurence) of event .
The conditional probability of given , written as , is defined by
If events and are independent, . Hence
In other words, the probability of given is just the probability of (regardless of — whether has occured or not doesn't affect the probability of , because they are independent events).
In a certain region, on any day of the year, the probability that it's cloudy is 0.4. It's also known that there's a 0.3 probability that any day is a cloudy and rainy day. Given that today is cloudy, what is the probability that it will rain?
Let be the event that it's cloudy and be the event that it rains.
Given today is cloudy, there's a 75% chance it will rain today.
In a city, the ratio between men and women is 6:4. Thirty percent of the men are vegetarian. What percentage of the city residents are vegetarian men?
Let be the probability of any random resident being a man and be the probability that any random resident being a vegetarian.
Eighteen percent of the city residents are vegetarian men.