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Additive and Multiplicative Rules for Probability

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Basic Probability

Probability is the study of chance. When studying probability, there are two very general classifications: theoretical probability and experimental probability .

  • Theoretical probability is the calculated probability that a given outcome will occur if the same experiment were completed an infinite number of times.
  • Experimental probability is the observed result of an experiment conducted a limited number of times.

The probability of a simple event is the calculated chance of a specific direct outcome of a single experiment where in all possible outcomes are equally likely. To calculate the probability of such an outcome, we use a very simple and intuitive formula:

Vocabulary

An event is something that occurs, or happens, with one or more possible outcomes.

An experiment is the process of taking a measurement or making an observation.

simple event is the simplest outcome of an experiment.

The sample space is the set of all possible outcomes of an experiment, typically denoted by \begin{align*}S\end{align*} .

To calculate the union, you just add up the individual probabilities of the events (Additive Rule of Probability).

For mutually exlusive events:

conditional probability formula :

\begin{align*}P(A|B)=\frac{P(A \cap B)}{P (B)}\end{align*}

This is read as "The probability that \begin{align*}A\end{align*} will occur, given that \begin{align*}B\end{align*} will occur (or has occurred), is equal to the intersection of probabilities \begin{align*}A\end{align*} and \begin{align*}B\end{align*} divided by the probability of \begin{align*}B\end{align*} alone".

Complement of an Event

The complement of an event is the sample space of all outcomes that are not the event in question.

Complements are notated using the prime symbol ’ as in: P(A') is the complement of P(A).

To calculate the probability of the complement of an event, use the following formula: P(A') = 1 - P(A)

Sometimes the probability of an event is difficult or impossible to calculate directly.  In this case, it may be easier to caclulate the probability of the complement of the event, and then subtract that from 1 to get the probability of the actual event.


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