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# Mutually Inclusive Events

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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:

$P(x)=\frac{\text{number of events where} \ x \ \text{is true}}{\text{total number of possible events}}$

Where $P(x)$ is the probability that $x$ will occur

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 $S$ .

Types of Events

mutually exclusive: two events that cannot occur at the same time

mutually inclusive: two events that can occur at the same time

independent: knowing the probability of one event does not affect the probability of the other

dependent: knowing the probability of one event does affect the probability of the other

Union of Compound Events

The union is the probability of any one of multiple events happening at a given time.  We denote the union of the two events by the symbol $A \cup B$ .

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

For mutually exlusive events:

P(x or y) = P(x) + P(y)

For mutually inclusive events:

P(x or y) = P(x) + P(y) - P(x and y)

Intersection of Compound Events

The intersection is the probability of both or all of the events you are calculating happening at the same time.  We denote the intersection of two events by the symbol $A \cap B$ .

To calculate the intersection, you multiply the individual probabilities of the events (Multiplicative Rule of Probability).

For independent events:

P(x and y) = P(x) * P(y)

For dependent events:

P(x and y) = P(x|y) * P(y)

Conditional Probability

We use conditional probability to talk about the probability that a certain event occurs given that another event has already occured.

conditional probability formula :

$P(A|B)=\frac{P(A \cap B)}{P (B)}$

This is read as "The probability that $A$ will occur, given that $B$ will occur (or has occurred), is equal to the intersection of probabilities $A$ and $B$ divided by the probability of $B$ 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.