1.5 Conditional Probability

1. Conditional Probability

2. The multiplication rule

3. Partition Theorem

4. Bayes’ Rule

1.5.1 Conditional Probability

Conditioning is another of the fundamental tools of probability: probably the most fundamental tool. It is especially helpful for calculating the probabilities of intersections, such as P(A∩B).

Additionally, the whole field of stochastic processes随机过程 is based on the idea of conditional probability. What happens next in a process depends, or is conditional, on what has happened beforehand.

Dependent events

Suppose A and B are two events on the same sample space. There will often be dependence between A and B. This means that if we know that B has occurred, it changes our knowledge of the chance that A will occur.

Example: Toss a dice once.

Conditioning as reducing the sample space

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Definition of conditional probability

Conditional probability provides us with a way to reason about the outcome of an experiment based on partial information.

Conditional Probabilities Satisfy the Three Axioms

Conditional Probabilities Satisfy General Probability Laws

Simple Example using Conditional Probabilities

1.5.2 The Multiplication Rule

The multiplication rule is most useful when the experiment consists of several stages in succession. The conditioning event B then describes the outcomes of first stage and A is the outcome of the second. So that P(A|B) –conditioning on what occurs first –will often be known. The rule can be easily extended to experiments involving more than two stages.

Example Three cards are drawn from an ordinary 52-card deck without replacement (drawn cards are not placed back in the deck). We wish to find the probability that none of the three cards is a “heart”.

A Partition of the sample space

Examples:

Partitioning an event A

The Partition Theorem

The Partition Theorem.

Total Probability Formula

Example Using Total Probability Theorem

You enter a chess tournament where your probability of winning a game is 0.3 against half the players (call them type 1),0.4 against a quarter of the player (call them type 2), You play a game against a randomly chosen opponent. What is the probability of winning?

Example 1.18 P15

In a certain assembly plant, three machines, B1,B2, and B3, make 30%,45%,and 25%, respectively ,of the products. It is known from past experience that 2%,3% and 2% of the products made by each machine, respectively , are defective次品 . Now, suppose that a finished product is randomly selected. What is the probability that it is defective?

Solution Let A=“the product is defective”,

Bi=“the product is made by machine Bi”.

1.5.4 Bayes’ Theorem

Example 1.19 P17

With reference to Example 1.18, if a product were chosen randomly and found to be defective, what is the probability that it was made by machine B3?

Example The False-Positive Puzzle