# Ch3: Conditional probability and independence

## Conditional probability

Conditional probability is the probability of one event occurring with some relationship to one or more other events.

Definition conditional probablity

When

as the (conditional) probability of

The following Multiplication rule follows:

Muliplication Rule

Independent Events

Dependent Events

## Law of total probability and Bayes' rule

The law of total probability states that If

Bayes' info

Note that the Bayes' info can be used in combination with the multiplication rules for example with dependent events:

## Independence of events and random variables

Independence

The events 𝐴 and 𝐵 are **independent** when:

Bernoulli trials

A series of experiments is called Bernoulli experiments or trials if

- each experiment has two possible outcomes, often denoted with 'Success' and 'Failure',
- the experiments are independent and
- the probability of success is the same for each experiment.

From this follows the **Binomial formula** If

If we conduct Bernoulli trials with success probability

Remember that $$p^{k}(1-p)^{n-k}$$ is the probability that the first