Stat 225: Topic 3 Conditional Probability and Independence
Conditional Probability
Suppose we conduct an experiment in which event Aand event Bare possible results, and
Aand Bare not necessarily mutually exclusive. Let’s suppose that we can quickly find out
the results of event A, but event Btakes more time to determine (it might be a future part
of a multi-step experiment, not revealed immediately, etc.) Sometimes, knowing whether or
not Ahappened can give us more insight into how likely it is that Bhappened - before we
actually learn the results of B. This is known as a conditional probability, and it comes up
quite often, sometimes with surprising results. Some applications include:
•Disease diagnosis - some diseases cannot be diagnosed with certainty until after a
person has died. When one of these diseases is suspected, the patient will often be
given a screening test which is not 100% accurate. Event Awould consist of a positive
screening test result, and event Bwould be actually having the disease.
•Market prediction - before companies launch new products, they will often test them
out in test markets. The purpose of this is to determine how likely the product is to be
successful before investing a large amount of money to produce thousands or millions
of copies of the product.
•Sports game predictions - you may not realize it, but when you predict the outcome
of the next game, you’re using conditional probability, at least at some level. You’re
prediction of what will happen in game xis based on what has happened in all the
previous games.
Definition: If Aand Bare events, then the conditional probability of Bgiven Ais
Remark: This is a lot easier to see using a Venn Diagram:
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