The conditional probability of an event A, given an event B with:Is defined by:
This last equation specifies a new (conditional) probability law on the same sample space Ω. In particular, all properties of probability laws remain valid for conditional probability laws.
- Conditional probabilities can also be viewed as a probability law on a new universe B, because all of the conditional probability is concentrated on B.
- If the possible outcomes are finitely many and equally likely, then:
Conditional probability provides us with a way to reason about the outcome of an experiment, based on partial information. Here are some examples of situations we have in mind:
- In an experiment involving two successive rolls of a die, you are told that the sum of the two rolls is 9. How likely is it that the first roll was a 6?
- In a word guessing game, the first letter of the word is a “t”. What is the likelihood that the second letter is “h”?
- How likely is it that a person has a certain disease given that a medical test was negative?
- A spot shows up on a radar screen. How likely is it to correspond to an aircraft?
In more precise terms, given an experiment, corresponding sample spaces, and a probability law, suppose that we know that the outcome is within some given event B. We wish to quantify the likelihood that the outcome also belongs to some other given event A. We thus seek to construct a new probability law that takes into account the available knowledge: a probability law that for any event A, specifies the conditional probability of A given B, denoted by P(AIB).
An appropriate definition of conditional probability when all outcomes are equally likely is given by:
Generalizing the argument, we introduce the following definition of conditional probability:Where we assume that:
The conditional probability is undefined if conditioning event has zero probability. In words, out of the total probability of the elements of B, P(AIB) is the fraction that is assigned to possible outcomes that also belong to A.
- Introduction to probability (bertsekas, 2nd, 2008)
- Probability – The Science of Uncertainty and Data (MITx – 6.431x)
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